Philip Hans Franses
email: franses@few.eur.nl

Dick Van Dijk

Henk Kranendonk

C53 -Forecasting and Other Model Applications < C5 -Econometric Modeling < C -Mathematical and Quantitative Methods

Keywords: Model selection, Forecast combination Forecast combination, Model selection Forecast combination, Model selection

Model selection for forecast combination

Introduction

Forecast combination is nowadays seen as a useful tool in practical forecasting; see Bates and Granger (1969) for the initial idea and see Clemen (1989) and Timmermann (2006) for surveys. What is usually done in practice is to line up a range of possibly suitable models, see if these models perform well enough in an evaluation sample, and use the forecasts from these models in a hold-out sample to see which combination of the forecasts is best for the yet unseen forecast sample.

One may wonder however if the in-sample evaluation is that much important, knowing that one will combine the forecasts anyway. Indeed, one may expect that irrelevant or inadequate models may turn up to have little or no weight in the combination, and that their inadequacy becomes apparent. The question is now whether looking at those weights, which are typically obtained through auxiliary least-squares based regressions 1 , is informative enough. For example, it can happen that some weights are negative when forecasts are all on the same side of the true data points. The hold-out sample may also not be large enough to find significant weights. And, indeed, it may well be that even a small weight in the forecast combination can still be enough to establish better forecast performance.

In this paper I therefore recommend another model selection strategy, which is related to the notion of encompassing. That is, a model is selected in the final combination if the combination with that model yields more forecast accuracy than a combination without that model. For the sake of convenience I choose to look at mean squared prediction errors, but other criteria can be used as well. It is shown that, depending on the empirical setting, the test can be non-standard, and one then needs to follow the methodology outlined in Clark and McCracken (2001). An illustration to realtime forecasts for quarterly growth in Gross Domestic Product (GDP) in the Netherlands shows that useful conclusions can be drawn. A concluding section highlights a few issues for further research. 1 Examples of regression-based selection methods to examine which models should be included in the combined forecast are discussed in Harvey and Newbold (2000) and Swanson and Zeng (2001).

The main idea

In this section, I outline the main idea of the proposed model selection criterion, and I discuss a proper test statistic.

Preliminaries

Consider a time-series variable y and assume for the moment that there are two linear regression models to explain the variation in y, that is,

(1) M 1 :

1 1 1 + = X y and (2) M 2 : 2 2 2 + = X y
where there are k 1 regressors in M 1 and k 2 regressors in M 2 , each containing an intercept, and where parts of these regressors can overlap. For the moment it is assumed that the models are not nested, and below I will examine the consequence of relaxing this assumption. Further, the viewpoint is that an analyst starts from either M 1 or M 2 and that he or she wonders whether it is worthwhile to include the other model in a subsequent forecast combination.

Assume that the analyst has data for N = R + P observations, of which the first R are used to estimate the parameters of M 1 and M 2 , and the second set P is used to evaluate the quality of the one-step-ahead forecasts made from the two models.

Eventually, a one-step-ahead forecast for N + 1 has to be made.

It is assumed that the analyst has the intention to use a combined forecast for N + 1 and starting from M 1, he or she wonders whether M 2 should be considered in that combination or the other way around. So, each model delivers a forecast for that also because it is well known that combined forecasts usually perform better, see Clemen (1989) and Timmermann (2006).

The weights 1 and 2 are determined using the P hold-out observations. For that sample, the analyst has 2 times the P one-step-ahead forecasts, that is, (4)

P R R y y + + , 1 1 , 1 ,..., ˆand P R R y y + + , 2 1 , 2
,..., ˆ, from models M 1 and M 2 , respectively, with forecast errors (5

) P R R + + , 1 1 , 1 ,..., ˆ and P R R + + , 2 1 , 2 ,..., ˆ
The analyst uses the forecasts in (4) to estimate the weights 1 and 2 for (6)

t t t c y y y , 2 2 , 1 1 , ˆ + =
where the auxiliary regression to get these weights has t running from R + 1 to R + P.

Note that in practice P may be quite small. This combined forecast has one-step-ahead forecast errors

P R c R c + + , 1 , ,..., ˆ
. Timmermann (2006) gives a summary of useful methods to estimate these weights. When the variances of the one-step-ahead forecast errors are approximately equal, it often is found that the optimal weights are 0.5 and 0.5, or in general 1/K where K would be the number of models like in (1) and (2). For the moment, it suffices to assume that the analyst either fixes the weights a priori or (somehow)

estimates them using least squares.

Finally, it is assumed that the analyst repeats this way of combining forecasts each approaches 0, see Clark and McCracken (2001) for the terminology, which will become more relevant later. Below I shall discuss what happens to the test below when this does not approach 0.

Testing the relevance of a model

The main idea is that I would argue that a model should be selected and be included in the combined forecast if the final combined forecast is better off in terms of forecast accuracy with that particular model than without it.

To examine this, one could now consider the two encompassing regression models (8)

t t i t c t y y y + + + = , 2 , 1 0
ŵhere i is either 1 or 2 to indicate one of the models, and where t runs from R+1 to R+P, and to see if 2 =0. For example, when i = 1, and 2 =0, then model M 2 significantly adds to the combined forecast's accuracy. However, simulations in Clark and McCracken (2001), among others, have shown that such encompassing tests do not have much power.

Therefore, I propose to evaluate the actual forecasts using the test proposed in Ericsson (1992), which has proved its usefulness, see again the simulations in Clark and McCracken (2001), that is, the t-ratio of i in (9) Clark and McCracken (2001, pp. 91-92) show that this t-test has a standard normal distribution under the null hypothesis of equal forecast accuracy. As the alternative hypothesis is a one-sided hypothesis, that is, as the alternative is that the combined forecast is better, the 95% critical value of this test is 1.645.

When the null hypothesis is rejected for model M i , then the accuracy of the combined forecast is higher than that of M i alone and hence M i can be improved by adding information from the other model. If the null hypothesis is not rejected, M i 's forecast is equally good as the combined forecast and the other forecast (and model) does not matter, and there is then also no need to include it in the combined forecast.

Variations

There are two directions in which the above approach may need to be expanded to meet relevant practical situations. The first is that 0 and the second is that there are K > 2 models instead of just 2.

When 0 this means that if new data become available, that then P and R both increase such that approximately approaches a fixed constant. In that case it also becomes important to recognize that the combined forecast nests the individual forecasts, see Clark and McCracken (2001). In Clark and McCracken (2001) it is shown that the tratio for i in (9) then has no standard normal distribution anymore in the case of nested forecast schemes. Critical values for the cases where is 0.1, 0.2, 0.4, 1.0, 2.0, 3.0 and 5.0 are given in the first panel of their Table 1 (page 92).

The combined forecast can cover K models. Assuming that one looks at the situation where an earlier combination covering K-1 models can be improved with one additional model, the degree of nesting is 1.

An illustration

To illustrate this proposal for model selection, and also to illustrate how results can be interpreted, I consider the real-time forecasts in Table 1, and the realizations as they are presently known (May 26, 2008). I also report the flash values that were published six weeks after the relevant quarter, that is, the first-release data.

Currently available GDP data

The data concern annual growth rates of Netherlands' GDP, when predicted and observed for the quarters 2004Q4 to and including 2007Q4. This is not very large sample, and this is due to the fact that there are only two real-time forecasts available for the Netherlands, that is, the forecasts in the last two columns were published in the very same quarters as they concern.

Insert Table 1 about here

The second column of Table 1 contains the currently known GDP growth rates (computed as log(y)-log(y -4 ) where y is GDP in billions of euros), as they are published by Statistics Netherlands (CBS). The fourth column contains the real-time forecasts made by Consensus Economics Inc, a commercial London UK-based company that publishes real-time forecasts for many countries around the world, including the Netherlands.

These forecasts are based on weighted expert opinions, and details of the procedure are given on the website www.consensuseconomics.com. The final column contains the realtime forecasts created using the methodology in De Groot and Franses (2005)

µ + + = ) ( ˆ, ,
, one gets a t-ratio of 2.183. From (10b) it can thus be learned that the EICIE can be improved by including the Consensus forecast in the combination, while (10a) tells us that this equal-weights combination is not significantly better than the Consensus forecast already is.

One possible reason for the above finding is that a combined forecast based on other than equal weights is perhaps better. When I regress the CBS data on an intercept, the Consensus and EICIE forecasts I get as a combined forecast 1.257 + 0.429 Consensus + 0.215 EICIE with an R 2 value of 0.541. The t-ratios of these parameters do not indicate significance, which is of course due to the small sample size of just 13 observations. This emphasizes the problems of interpreting the t-ratios of parameters in combining regressions, and it reiterates the very reason to look at test statistics like those in (10a) and (10b). Note that the estimated intercept takes a large positive value, and this reflects the commonly found phenomenon that later vintages of data typically move upwards, relative to the first release (flash) values. When the intercept is not included, the combined forecast is 0.851 Consensus + 0.285 EICIE where the parameter for the Consensus forecast is significant at the 5% level, while that of the EICIE is not. 2 gives the forecast errors of this least-squaresweights-combined forecast with an intercept included, while the last column gives it for the case without an intercept. Clearly, this combined forecast outperforms its constituents by far, and especially the mean squared prediction error is reduced substantially.

To verify if each of the two models is contributing significantly or whether one of the component forecasts is better on its own, I run the regression for the case of the intercept included and get (11a)

t t resweights leasstsqua combined t consensus t consensus µ + + = ) ( ˆ, ,
, with a t-ratio for equal to 2.162, while the t-ratio for in (11b)

t t resweights leasstsqua combined t eicie t eicie µ + + = ) ( ˆ, ,
, is equal to 3.016. Hence, now both models are relevant for the combined forecast, albeit that the EICIE needs less weight than the Consensus forecast in this final combined forecast. The conclusion here is that both individual real time forecasts for Netherlands GDP can be improved by combining them with the other in a linear combination which has no equal weights. In case the least-squares-weights combined forecast does not include an intercept, the t-ratios for parameters in (11a) and (11b) are -0.107 and 1.618, respectively, which shows that the intercept was needed indeed.

Overall, these results show that for the final value of real GDP growth a combination of the Consensus and EICIE forecasts can be beneficial, where the contribution of the EICIE is relatively smaller than that of the Consensus. Both forecasts seem to underestimate the final value, and hence the combination requires a non-zero intercept to accommodate for this.

First-release GDP data

To see if the results for the final data carry trough to the first-release data, I consider the flash values of GDP. The third column of Table 1 gives the first-release growth rates, as they were published about six weeks after the end of the relevant quarter. Comparing the numbers in columns 2 and 3 of Table 1, one can see that there can be substantial differences between first-release and the currently seen as "final" values. This can also be observed from the graphs in Figures 2 and3.

Insert Figures 2 and 3 about here

Insert Table 3 about here Columns 2 and 3 of Table 3 show that both the Consensus and EICIE forecasts do much better on the flash data, where again the Consensus forecasts outperform.

To see if either the Consensus forecasts or the EICIE forecasts can be improved by including the other, let us again look at a combination based on equal (0.5) weights.

The forecast errors of this new forecast are displayed in the column "Equal weights" of Table 3. As expected, the mean and median errors of this equal weight forecast combination are in between those of the two forecasts. The mean squared prediction error (0.66) is closer but larger than that (0.56) of the Consensus forecast, while the median squared forecast error (0.06) is substantially smaller than each of its components. Hence, equal weights certainly do give some improvement.

The first test regression assumes that the Consensus forecasts are the starting point and it looks at whether it can be improved by incorporating the EICIE forecasts. It reads as (10a) and it gives a t-ratio for of 0.496, while for the test regression (10b) one gets a t-ratio of 3.011. From (10b) it can thus be leaned that the EICIE can be improved by including the Consensus forecast in the combination, while (10a) tells us that this equal-weights combination is not significantly better than the Consensus forecast. with an R 2 value of 0.649, where now the intercept is considerably smaller than before, and it is now found not statistically significant. When the intercept is not included, the combined forecast is 0.891 Consensus + 0.110 EICIE where the parameter for the Consensus forecast is significant at the 5% level, while that of the EICIE is not. Note that the sum of the parameters is about equal to 1.

The penultimate column of Table 3 gives the forecast errors of this least-squaresweights-combined forecast with an intercept included, while the last column gives it for the case without an intercept. Clearly, this combined forecast outperforms its constituents by far, and especially the median squared prediction error is reduced substantially.

Running regression (11a) for the case of the intercept included I get a t-ratio for equal to 0.837, while the t-ratio for in (11b) is equal to 3.143. In case the least-squaresweights combined forecast does not include an intercept, the t-ratios for parameters in (11a) and (11b) are 0.488 and 3.009, respectively. These results show that for the flash value of real GDP growth a combination of the Consensus and EICIE forecasts is not beneficial, as the contribution of the EICIE is not significant relative to that of the Consensus. Hence there only the Consensus forecasts will do. As we saw, for the revised ("final") GDP figures, the EICIE does seem to be relevant for the final combination.

Consequences

This paper has put forward a simple methodology to see if forecasts from models can be significantly improved by combining them with forecasts from other models. It has a single model or perhaps already a combination of K-1 forecasts as the starting points, and Looking at the use of model forecasts this way, in-sample model diagnostics have become less relevant. What is needed is a set of consistent forecasts from 2 or K models.

Moreover, forecasts in the past do not need to be accurate. Individual track records are no guarantee that the models will successfully contribute to the combined forecasts, so if the intention is to consider combined forecasts anyway, only studying in-sample performance becomes obsolete.

Future work on the issue of this paper could include model selection for the combination of multi-step-ahead forecasts. Also, many more examples would be needed to illustrate the merits.

Introduction

Forecast combination is nowadays seen as a useful tool in practical forecasting; see Bates and Granger (1969) for the initial idea and see Clemen (1989) and Timmermann (2006) for surveys. What is usually done in practice is to line up a range of possibly suitable models, to see if these models perform well enough in an evaluation sample, and to use the forecasts from these models in a hold-out sample to see which combination of the forecasts is best for the yet unseen forecast sample.

One may wonder however if the in-sample evaluation is that much important, knowing that one will combine the forecasts anyway. Indeed, one may expect that irrelevant or inadequate models may turn up to have little or no weight in the combination, and that their inadequacy becomes apparent. The question is now whether looking at those weights, which are typically obtained through auxiliary least-squares based regressions 1 , is informative enough. For example, it can happen that some weights are negative when forecasts are all on the same side of the true data points. The hold-out sample may also not be large enough to find significant weights. And, indeed, it may well be that even a small weight in the forecast combination can still be enough to establish better forecast performance.

In this paper it is therefore recommended to consider another model selection strategy, which is related to the notion of encompassing. That is, a model is selected in the final combination if the combination with that model yields more forecast accuracy than a combination without that model. For the sake of convenience only mean squared prediction errors are considered, but other criteria can be used as well. It is shown that, depending on the empirical setting, the test can be non-standard, and one then needs to follow the methodology outlined in Clark and McCracken (2001). An illustration to realtime forecasts for quarterly growth in Gross Domestic Product (GDP) in the Netherlands shows that useful conclusions can be drawn. A concluding section highlights a few issues for further research.

1 Examples of regression-based selection methods to examine which models should be included in the combined forecast are discussed in Harvey and Newbold (2000) and Swanson and Zeng (2001).

The main idea

In this section, the main idea of the proposed model selection criterion is outlined, and a proper test statistic is discussed.

Preliminaries

Consider a time-series variable y and assume for the moment that there are two linear regression models to explain the variation in y, that is,

(1) M 1 :

1 1 1 ε β + = X y and (2) M 2 : 2 2 2 ε β + = X y
where there are k 1 regressors in M 1 and k 2 regressors in M 2 , each containing an intercept, and where parts of these regressors can overlap. For the moment it is assumed that the models are not nested, but below the consequence of relaxing this assumption will be examined. Further, the viewpoint is that an analyst starts from either M 1 or M 2 and that he or she wonders whether it is worthwhile to include the other model in a subsequent forecast combination.

Assume that the analyst has data for N = R + P observations, of which the first R are used to estimate the parameters of M 1 and M 2 , and the second set P is used to evaluate the quality of the one-step-ahead forecasts made from the two models.

Eventually, a one-step-ahead forecast for N + 1 has to be made.

It is assumed that the analyst has the intention to use a combined forecast for N + 1 and starting from M 1, he or she wonders whether M 2 should be considered in that combination or the other way around. So, each model delivers a forecast for that α α also because it is well known that combined forecasts usually perform better, see Clemen (1989) and Timmermann (2006).

The weights α 1 and α 2 are determined using the P hold-out observations. For that sample, the analyst has 2 times the P one-step-ahead forecasts, that is, (4)

P R R y y + + , 1 1 , 1 ,..., ˆ and P R R y y + + , 2 1 , 2 ,..., ˆ,
from models M 1 and M 2 , respectively, with forecast errors (5

) P R R + + , 1 1 , 1 ,..., ˆε ε and P R R + + , 2 1 , 2 ,..., ˆε ε
The analyst uses the forecasts in (4) to estimate the weights α 1 and α 2 for (6)

t t t c y y y , 2 2 , 1 1 , α α + =
where the auxiliary regression to get these weights has t running from R + 1 to R + P.

Note that in practice P may be quite small. This combined forecast has one-step-ahead forecast errors

P R c R c + + , 1 ,
,..., ˆε ε . Timmermann (2006) gives a summary of useful methods to estimate these weights. When the variances of the one-step-ahead forecast errors are approximately equal, it often is found that the optimal weights are 0.5 and 0.5, or in general 1/K where K would be the number of models like in (1) and (2). For the moment, it suffices to assume that the analyst either fixes the weights a priori or (somehow) estimates them using least squares.

Finally, it is assumed that the analyst repeats this way of combining forecasts each time a new time-series observation becomes available. When this happens, the analyst 7) R P = π approaches 0, see Clark and McCracken (2001) for the terminology, which will become more relevant later. Below it will be discussed what happens to the test below when this π does not approach 0.

Testing the relevance of a model

The main idea is that it can be argued that a model should be selected and be included in the combined forecast if the final combined forecast is better off in terms of forecast accuracy with that particular model than without it.

To examine this, one could now consider the two encompassing regression models (8)

t t i t c t y y y ε β β β + + + = , 2 , 1 0 ˆ
where i is either 1 or 2 to indicate one of the models, and where t runs from R+1 to R+P, and to see if β 2 =0. For example, when i = 1, and β 2 =0, then model M 2 significantly adds to the combined forecast's accuracy. However, simulations in Clark and McCracken (2001), among others, have shown that such encompassing tests do not have much power. Therefore, in this paper it is proposed to evaluate the actual forecasts using the test proposed in Ericsson (1992), which has proved its usefulness, see again the simulations in Clark and McCracken (2001), that is, the t-ratio of α i in (9) Clark and McCracken (2001, pp. 91-92) show that this t-test has a standard normal distribution under the null hypothesis of equal forecast accuracy. As the alternative hypothesis is a one-sided hypothesis, that is, as the alternative is that the combined forecast is better, the 95% critical value of this test is 1.645.

t t c t i i t i η ε ε α ε + - = ) ( ˆ, , , F o r P e e r R e v i e w 6 When π = 0,
When the null hypothesis is rejected for model M i , then the accuracy of the combined forecast is higher than that of M i alone and hence M i can be improved by adding information from the other model. If the null hypothesis is not rejected, M i 's forecast is equally good as the combined forecast and the other forecast (and model) does not matter, and there is then also no need to include it in the combined forecast.

Variations

There are two directions in which the above approach may need to be expanded to meet relevant practical situations. The first is that π ≠ 0 and the second is that there are K > 2 models instead of just 2.

When π ≠ 0 this means that if new data become available, that then P and R both increase such that π approximately approaches a fixed constant. In that case it also becomes important to recognize that the combined forecast nests the individual forecasts, see Clark and McCracken (2001). In Clark and McCracken (2001) it is shown that the tratio for α i in (9) then has no standard normal distribution anymore in the case of nested forecast schemes. Critical values for the cases where π is 0.1, 0.2, 0.4, 1.0, 2.0, 3.0 and 5.0 are given in the first panel of their Table 1 (page 92).

The combined forecast can cover K models. Assuming that one looks at the situation where an earlier combination covering K-1 models can be improved with one additional model, the degree of nesting is 1.

An illustration

To illustrate this proposal for model selection, and also to illustrate how results can be interpreted, consider the real-time forecasts in Table 1, and the realizations as they are presently known (May 26, 2008). The flash values that were published six weeks after the relevant quarter, that is, the first-release data are also reported.

Currently available GDP data

The data concern annual growth rates of Netherlands' GDP, when predicted and observed for the quarters 2004Q4 to and including 2007Q4. This is not very large sample, and this is due to the fact that there are only two real-time forecasts available for the Netherlands, that is, the forecasts in the last two columns were published in the very same quarters as they concern.

Insert Table 1 about here The second column of Table 1 contains the currently known GDP growth rates (computed as log(y)-log(y -4 ) where y is GDP in billions of euros), as they are published by Statistics Netherlands (CBS). The fourth column contains the real-time forecasts made by Consensus Economics Inc, a commercial London UK-based company that publishes real-time forecasts for many countries around the world, including the Netherlands.

These forecasts are based on weighted expert opinions, and details of the procedure are given on the website www.consensuseconomics.com. The final column contains the realtime forecasts created using the methodology in De Groot and Franses (2005). This socalled Econometric Institute Current Indicator of the Economy (EICIE) is four-weekly published in the (Dutch language) Economische Statistische Berichten. The graphs in Figure 1 show that the two forecasts seem to follow the actual GDP values reasonably well. Some observations are predicted rather well, but sometimes the fit is poor. The real-time forecasts in the EICIE are based on a regression of GDP (after suitable transformation) on current and lagged growth rates of employees in the temporary staffing sector and on its own past This autoregressive distributed lag model, when estimated in error correction format to incorporate a cointegration relation, is re-estimated each time a new observation becomes available. Hence, the size of P is 1, and R increases each time with 1, which makes π to approach 0. It is not evident whether the value of π is also approximately equal to 0 for the Consensus forecasts, but it seems a reasonable assumption too.

Insert Table 2 about here

Table 2 gives the one-step-ahead (or better: current) forecast errors for the two forecasts in columns 4 (Consensus) and 5 (EICIE) of Table 1. The mean error, median error and mean squared prediction error of Consensus are smaller than those of the EICIE, and the EICIE has a smaller median squared prediction error. Based on these two individual track records, one would be inclined to favor the Consensus forecasts.

As said, it is assumed that one aims at considering a combined forecast, so now it matters if either the Consensus forecasts or the EICIE forecasts can be improved by including the other. Let us first look at a combination based on equal (0.5) weights. The forecast errors of this new forecast are displayed in the fourth column of Table 2. As expected, the mean and median errors of this equal weight forecast combination are in between those of the two forecasts. The mean squared prediction error (1.01) is closer but larger than that (0.93) of the Consensus forecast, while the median squared forecast error (0.25) is smaller than each of its components. Hence, equal weights do give some improvement, but not that much. This result is emphasized by the outcomes of the test regression in (9). The first regression assumes that the Consensus forecasts are the starting point and it looks at

η ε ε α µ ε + - + = - ) ( ˆ, ,
, one gets a t-ratio of 2.183. From (10b) it can thus be learned that the EICIE can be improved by including the Consensus forecast in the combination, while (10a) tells us that this equal-weights combination is not significantly better than the Consensus forecast already is.

One possible reason for the above finding is that a combined forecast based on other than equal weights is perhaps better. When the CBS data are regressed on an intercept, the Consensus and EICIE forecasts the combined forecast becomes 1.257 + 0.429 Consensus + 0.215 EICIE with an R 2 value of 0.541. The t-ratios of these parameters do not indicate significance, which is of course due to the small sample size of just 13 observations. This emphasizes the problems of interpreting the t-ratios of parameters in combining regressions, and it reiterates the very reason to look at test statistics like those in (10a) and (10b). Note that the estimated intercept takes a large positive value, and this reflects the commonly found phenomenon that later vintages of data typically move upwards, relative to the first release (flash) values. When the intercept is not included, the combined forecast is 0.851 Consensus + 0.285 EICIE where the parameter for the Consensus forecast is significant at the 5% level, while that of the EICIE is not. 2 gives the forecast errors of this least-squaresweights-combined forecast with an intercept included, while the last column gives it for the case without an intercept. Clearly, this combined forecast outperforms its constituents by far, and especially the mean squared prediction error is reduced substantially.

To verify if each of the two models is contributing significantly or whether one of the component forecasts is better on its own, one can run the regression for the case of the intercept included and get (11a)

t t resweights leasstsqua combined t consensus t consensus η ε ε α µ ε + - + = - ) ( ˆ, ,
, with a t-ratio for α equal to 2.162, while the t-ratio for α in (11b)

t t resweights leasstsqua combined t eicie t eicie η ε ε α µ ε + - + = - ) ( ˆ, ,
, is equal to 3.016. Hence, now both models are relevant for the combined forecast, albeit that the EICIE needs less weight than the Consensus forecast in this final combined forecast. The conclusion here is that both individual real time forecasts for Netherlands GDP can be improved by combining them with the other in a linear combination which has no equal weights. In case the least-squares-weights combined forecast does not include an intercept, the t-ratios for α parameters in (11a) and (11b) are -0.107 and 1.618, respectively, which shows that the intercept was needed indeed.

Overall, these results show that for the final value of real GDP growth a combination of the Consensus and EICIE forecasts can be beneficial, where the contribution of the EICIE is relatively smaller than that of the Consensus. Both forecasts seem to underestimate the final value, and hence the combination requires a non-zero intercept to accommodate for this.

First-release GDP data

To see if the results for the final data carry trough to the first-release data, consider the flash values of GDP. The third column of Table 1 gives the first-release growth rates, as they were published about six weeks after the end of the relevant quarter. Comparing the numbers in columns 2 and 3 of Table 1, one can see that there can be substantial differences between first-release and the currently seen as "final" values. This can also be observed from the graphs in Figures 2 and3.

Insert Figures 2 and 3 about here

Insert Table 3 about here Columns 2 and 3 of Table 3 show that both the Consensus and EICIE forecasts do much better on the flash data, where again the Consensus forecasts outperform.

To see if either the Consensus forecasts or the EICIE forecasts can be improved by including the other, let us again look at a combination based on equal (0.5) weights.

The forecast errors of this new forecast are displayed in the column "Equal weights" of Table 3. As expected, the mean and median errors of this equal weight forecast combination are in between those of the two forecasts. The mean squared prediction error (0.66) is closer but larger than that (0.56) of the Consensus forecast, while the median squared forecast error (0.06) is substantially smaller than each of its components. Hence, equal weights certainly do give some improvement.

The first test regression assumes that the Consensus forecasts are the starting point and it looks at whether it can be improved by incorporating the EICIE forecasts. It reads as (10a) and it gives a t-ratio for α of 0.496, while for the test regression (10b) one gets a t-ratio of 3.011. From (10b) it can thus be leaned that the EICIE can be improved by including the Consensus forecast in the combination, while (10a) tells us that this equal-weights combination is not significantly better than the Consensus forecast.

When the CBS data are regressed on an intercept, the Consensus and EICIE forecasts one gets a combined forecast where the parameter for the Consensus forecast is significant at the 5% level, while that of the EICIE is not. Note that the sum of the parameters is about equal to 1.

The penultimate column of Table 3 gives the forecast errors of this least-squaresweights-combined forecast with an intercept included, while the last column gives it for the case without an intercept. Clearly, this combined forecast outperforms its constituents by far, and especially the median squared prediction error is reduced substantially.

Running regression (11a) for the case of the intercept included gives a t-ratio for α equal to 0.837, while the t-ratio for α in (11b) is equal to 3.143. In case the least-squaresweights combined forecast does not include an intercept, the t-ratios for α parameters in (11a) and (11b) are 0.488 and 3.009, respectively. These results show that for the flash value of real GDP growth a combination of the Consensus and EICIE forecasts is not beneficial, as the contribution of the EICIE is not significant relative to that of the Consensus. Hence there only the Consensus forecasts will do. As we saw, for the revised ("final") GDP figures, the EICIE does seem to be relevant for the final combination.

Consequences

This paper has put forward a simple methodology to see if forecasts from models can be significantly improved by combining them with forecasts from other models. It has a single model or perhaps already a combination of K-1 forecasts as the starting points, and Looking at the use of model forecasts this way, in-sample model diagnostics have become less relevant. What is needed is a set of consistent forecasts from 2 or K models. Moreover, forecasts in the past do not need to be accurate. Individual track records are no guarantee that the models will successfully contribute to the combined forecasts, so if the intention is to consider combined forecasts anyway, only studying in-sample performance becomes obsolete.

Future work on the issue of this paper could include model selection for the combination of multi-step-ahead forecasts. Also, many more examples would be needed to illustrate the merits.

  When I regress the CBS data on an intercept, the Consensus and EICIE forecasts I get as a combined forecast

Figure 1 :

 1 Figure 1: The data from Table 1 (Final data)

  keep the size of the hold-out sample P fixed, and increase the model estimation sample R to R+1. Upon doing so, the ratio (

  needs to think for a moment about the possible value of π.

  used in case one wonders whether a further combination with yet a new model can yield even better forecasts.

Figure 1 :

 1 Figure 1: The data from Table 1 (Final data)

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	it can be used in case one wonders whether a further combination with yet a new model
	can yield even better forecasts.
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Table 1 (

 1 Final data) This column gives the data on annual growth rates of GDP in the Netherlands, as they are published by the Central Bureau of Statistics, and as they are published on the website www.cbs.nl on April 22 2008.This column gives the data on annual growth rates of GDP in the Netherlands, as they are published by the Central Bureau of Statistics, and as they are published on the website www.cbs.nl six weeks after the end of the relevant quarter. This column gives the forecasts as they are created by Consensus Economics Inc., using the method outlined on their website www.consensuseconomics.com 4This column gives the quotes of the Econometric Institute Current Indicator of the Economy, as they are published in the (Dutch language) Economische Statistische Berichten, and on the website www.esbonline.nl

	Submitted Manuscript

Page 14 of 42 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Page 15 of 42 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Page 16 of 42 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK 2 Page 17 of 42 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Page 18 of 42 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Table 2 :

 2 Forecast errors corresponding to individual forecasting models and to forecast combinations (final CBS values, as quoted on May 26 2008).

			Forecasts	Combined forecasts, weights
	Quarter	Consensus	EICIE	Equal	LS, intercept
					With Without
	2004Q4	1.5	1.6	1.55	0.68 1.37
	2005Q1	-0.2	-0.5	-0.35	-1.27 -0.38
	2005Q2	1.3	3.1	2.20	0.55 1.77
	2005Q3	1.5	0.4	0.95	0.19 1.12
	2005Q4	0.9	0.1	0.50	-0.18 0.54
	2006Q1	0.4	1.2	0.80	0.35 0.21
	2006Q2	0.3	0.6	0.45	0.04 0.00
	2006Q3	-0.1	0.9	0.40	-0.18 -0.21
	2006Q4	-0.7	0.4	-0.15	-0.59 -0.85
	2007Q1	-0.7	0.0	-0.35	-0.74 -0.94
	2007Q2	-0.4	-0.5	-0.45	-0.68 -0.84
	2007Q3	1.4	1.4	1.40	1.08 1.02
	2007Q4	1.3	1.0	1.15	1.04 0.78
	Mean error	0.50	0.75	0.62	0.02 0.28
	Median error	0.40	0.60	0.50	0.04 0.21
	Mean SPE	0.93	1.43	1.01	0.48 0.82
	Median SPE	0.49	0.36	0.25	0.38 0.70

Table 3 :

 3 Forecast errors corresponding to individual forecasting models and to forecast combinations (flash CBS values).

			Forecasts	Combined forecasts, weights
	Quarter	Consensus	EICIE	Equal	LS, intercept
					With Without
	2004Q4	0.1	0.2	0.15	-0.09 0.11
	2005Q1	-1.0	-1.3	-1.15	-1.29 -1.03
	2005Q2	1.0	2.8	1.90	0.84 1.20
	2005Q3	0.4	-0.7	-0.15	0.01 0.28
	2005Q4	0.6	-0.2	0.20	0.30 0.51
	2006Q1	-0.2	0.6	0.20	-0.08 -0.11
	2006Q2	-0.4	-0.1	-0.25	-0.37 -0.37
	2006Q3	-0.3	0.7	0.20	-0.19 -0.19
	2006Q4	-0.7	0.4	-0.15	-0.51 -0.58
	2007Q1	-0.7	0.0	-0.35	-0.58 -0.63
	2007Q2	-0.6	-0.7	-0.65	-0.58 -0.61
	2007Q3	1.3	1.3	1.30	1.31 1.30
	2007Q4	1.2	0.9	1.05	1.23 1.16
	Mean error	0.05	0.30	0.18	0.00 0.08
	Median error	-0.20	0.20	0.15	-0.09 -0.11
	Mean SPE	0.56	1.09	0.66	0.52 0.55
	Median SPE	0.36	0.49	0.06	0.27 0.34

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	F o
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	3

Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

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					Submitted Manuscript
	whether it can be improved by incorporating the EICIE forecasts in an equal-weight
	combination. It reads as
	(10a)	ts ˆ, , equalweigh combined t consensus t consensus ε ε α µ ε -+ = -( ,	t	)	+	η	t
	and this regression gives a t-ratio for α of 1.139, while for the test regression
	(10b)	eicie	t	eicie	t	combined	equalweigh	ts	t	t
		F o
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			P
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											9

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	0.359 + 0.779 Consensus + 0.090 EICIE
	with an R 2 value of 0.649, where now the intercept is considerably smaller than before,
	and it is now found not statistically significant. When the intercept is not included, the
	combined forecast is
	0.891 Consensus + 0.110 EICIE
	F o
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Table 1 (

 1 Final data)

	Page

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	5
	4
	3
	2
	1
	0
	-1	F o
	-2	r
		05Q1 CBSFLASH 05Q3 P e 06Q1 e CONSENSUS 06Q3 07Q1 r Figure 2: The data from Table 1 (First release data) 07Q3 EICIE
		R e v i e w
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  Figure 3: the two CBS series (First release data and final data, as of May 26 2008)

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	5
	4
	3
	2
	1
	0	F o F o
	-1		r r
		05Q1	P e e r 05Q3 06Q1 06Q3 CBSFLASH P e e r	07Q1 CBS	07Q3
			R e v i e w R e v i e w
					16 17

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Table 1 :

 1 The data used in the illustrationsThis column gives the data on annual growth rates of GDP in the Netherlands, as they are published by the Central Bureau of Statistics, and as they are published on the website www.cbs.nl on April 22 2008.This column gives the data on annual growth rates of GDP in the Netherlands, as they are published by the Central Bureau of Statistics, and as they are published on the website www.cbs.nl six weeks after the end of the relevant quarter. This column gives the forecasts as they are created by Consensus Economics Inc., using the method outlined on their website www.consensuseconomics.com 4This column gives the quotes of the Econometric Institute Current Indicator of the Economy, as they are published in the (Dutch language) Economische Statistische Berichten, and on the website www.esbonline.nl

	Forecasts

2 Page 38 of 42 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

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Table 2 :

 2 Forecast errors corresponding to individual forecasting models and to forecast combinations (final CBS values, as quoted on May 26 2008).

			Forecasts	Combined forecasts, weights
	Quarter	Consensus	EICIE	Equal	LS, intercept
					With Without
	2004Q4	1.5	1.6	1.55	0.68 1.37
	2005Q1	-0.2	-0.5	-0.35	-1.27 -0.38
	2005Q2	1.3	3.1	2.20	0.55 1.77
	2005Q3	1.5	0.4	0.95	0.19 1.12
	2005Q4	0.9	0.1	0.50	-0.18 0.54
	2006Q1	0.4	1.2	0.80	0.35 0.21
	2006Q2	0.3	0.6	0.45	0.04 0.00
	2006Q3	-0.1	0.9	0.40	-0.18 -0.21
	2006Q4	-0.7	0.4	-0.15	-0.59 -0.85
	2007Q1	-0.7	0.0	-0.35	-0.74 -0.94
	2007Q2	-0.4	-0.5	-0.45	-0.68 -0.84
	2007Q3	1.4	1.4	1.40	1.08 1.02
	2007Q4	1.3	1.0	1.15	1.04 0.78
	Mean error	0.50	0.75	0.62	0.02 0.28
	Median error	0.40	0.60	0.50	0.04 0.21
	Mean SPE	0.93	1.43	1.01	0.48 0.82
	Median SPE	0.49	0.36	0.25	0.38 0.70

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Table 3 :

 3 Forecast errors corresponding to individual forecasting models and to forecast combinations (flash CBS values).

			Forecasts	Combined forecasts, weights
	Quarter	Consensus	EICIE	Equal	LS, intercept
					With Without
	2004Q4	0.1	0.2	0.15	-0.09 0.11
	2005Q1	-1.0	-1.3	-1.15	-1.29 -1.03
	2005Q2	1.0	2.8	1.90	0.84 1.20
	2005Q3	0.4	-0.7	-0.15	0.01 0.28
	2005Q4	0.6	-0.2	0.20	0.30 0.51
	2006Q1	-0.2	0.6	0.20	-0.08 -0.11
	2006Q2	-0.4	-0.1	-0.25	-0.37 -0.37
	2006Q3	-0.3	0.7	0.20	-0.19 -0.19
	2006Q4	-0.7	0.4	-0.15	-0.51 -0.58
	2007Q1	-0.7	0.0	-0.35	-0.58 -0.63
	2007Q2	-0.6	-0.7	-0.65	-0.58 -0.61
	2007Q3	1.3	1.3	1.30	1.31 1.30
	2007Q4	1.2	0.9	1.05	1.23 1.16
	Mean error	0.05	0.30	0.18	0.00 0.08
	Median error	-0.20	0.20	0.15	-0.09 -0.11
	Mean SPE	0.56	1.09	0.66	0.52 0.55
	Median SPE	0.36	0.49	0.06	0.27 0.34

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript


Bram ; Van Dijk

Philip Hans Franses

Richard Paap

Dick Van Dijk
email: djvandijk@ese.eur.nl.

Philip Hans

Franses Richard

Paap Dick Van Dijk

Modeling Regional House Prices

Keywords: cross-section dependence, cointegration, ripple effect cross-section dependence, cointegration, ripple effect JEL Classification: C21, C23, C53

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Real estate prices in many countries have experienced a dramatic boom in recent years (IMF, 2004). At the same time, the extent of the price increase appears to vary substantially across different regions within a given country. In the Netherlands, for example, it is commonly believed that house prices in Amsterdam and the densely populated western part of the country have increased far more than prices in the smaller cities and rural areas in the east. As house prices are typically available per region or city, we may analyze these data at such a disaggregate level, to examine whether indeed regions or cities behave differently, perhaps in terms of trends, but also in terms of response to outside economic shocks. In this paper we develop a time series model that suits this purpose.

Most regional house prices have the following properties. First, they tend to display a trend, and historical price patterns suggest that this trend probably is not deterministic but stochastic. In particular, house prices show 'bubble'-type behavior, where prolonged periods of steady increases of the price level suddenly end with a sharp drop followed by a period of low price levels, suggesting that trends are unlikely to be deterministic. Second, for different regions within a country these stochastic trends should somehow be linked.

It is not plausible that prices in different regions would diverge indefinitely or that certain regions would not respond to common macroeconomic shocks. So, a model for regional house prices should allow for some form of common trends. Third, it can be expected that adjacent regions show similar price patterns, although this may also be the case for regions far apart geographically but with similar economic or demographic characteristics.

Hence, a suitable model should allow for similarities in the dynamic behavior of house prices across regions. An intuitively appealing possibility is to consider a model that allows for groups or clusters of regions, where house price dynamics in regions within a given cluster are the same, while they are different across clusters. Preferably, such a model should not require ex-ante or exogenous assignment of regions to specific clusters. In fact it would be best if the data themselves were allowed to indicate if clusters exist and if so, which regions belong to which cluster.

In this paper we extend the latent-class panel time series model introduced by [START_REF] Paap | Does Africa Grow Slower Than Asia, Latin America and the Middle East? Evidence from a New Data-Based Classification Method[END_REF] to capture these different properties of regional house prices. The key feature of this model is that the clustering of regions is purely data-driven, where cluster membership is based on characteristics corresponding to two specific research questions we want to address. The first question is whether prices in all regions have the same average growth rate. Note that a common trend specification across the regions entails that their growth rates must be somehow compatible, but it still leaves open the possibility that house prices in some regions grow faster than in others. The second question we consider is the way the house prices in each region react to changes in the overall economic situation, which we measure by GDP. We examine both the size of the effect from GDP on the house prices and the speed at which regions react to changes in GDP.

We apply our model to house price data for the Netherlands, comprising 76 regions for which we have quarterly data for the period 1985Q1-2005Q4. We find that the 76 regions can be grouped into two clusters. The first cluster consists mainly of regions in the east of the country. These are mainly rural areas that are close to the larger cities, especially close to the Randstad (consisting of Utrecht, Amsterdam, Den Haag, Rotterdam and other cities in the area). This cluster reacts both stronger and faster to changes in GDP. The average growth rate does not vary over the regions.

There are not many studies that describe regional house prices. [START_REF] Cameron | Was there a British House Price Bubble? Evidence from a Regional Panel[END_REF] build a model from inverse demand equations. They have, however, only a limited number (9) of regions, and their model would not work in our situation where we have many more (76) regions, as we will describe below. [START_REF] Malpezzi | A Simple Error Correction Model of House Prices[END_REF] constructs an error correction model for regional house prices. The parameters of this model are however not allowed to vary across regions. [START_REF] Holly | A Spatio-Temporal Model of House Prices in the US[END_REF] In this paper we cover the middle ground, that is, the model parameters are allowed to vary across groups of regions but not across each region individually.

Before we propose our latent-class model, we first provide some details on the house price data in Section 2. We consider two decades of quarterly house prices on 76 regions in the Netherlands. We discuss their trending behavior by performing panel unit root tests and we also show that the growth rates in different regions show strong cross-correlations.

Using multidimensional scaling techniques we get a first impression if and how these 76 regions could get clustered. Then, in Section 3, we put forward our model specification, highlighting the underlying data-driven clustering mechanism. In addition, we describe the method used for parameter estimation. In Section 4 we first present our estimation results, and give interpretation to the various outcomes. Next, we take a look at impulse response functions of the house prices with respect to a shock in GDP and in the interest rate. In Section 5 we conclude with some limitations and we outline topics for further research.

Data

The Dutch real estate agent association [NVM] publishes quarterly data on house prices for N = 76 regions in the Netherlands. Our dataset covers the sample period 1985Q1-2005Q4 (T = 84 quarters). Hence, we have a panel database where both the cross-section dimension N and the time dimension T are fairly large.

The way the country is divided into 76 regions is determined by the NVM. Macroeconomic data, such as output and inflation, are not available for this particular specification of regions. Other (macro) variables that we use in our model are therefore measured at the country level. In particular, this concerns the interest rate (obtained from the Dutch Central Bank) and quarterly real GDP (from Statistics Netherlands). The GDP series is available until 2005Q2. We obtain real house prices by deflating with the consumer price index [CPI] (from Statistics Netherlands). In addition, we seasonally adjust the real GDP series using the Census X-12 algorithm (available in EViews 5.1). We denote the real house price in region i at time t as p i,t , and real GDP as y t .

[Figure 1 about here.]

Figure 1 shows time series of log(p i,t ) for three specific regions: Noordwest-Friesland, which usually is the least expensive region, Bunnik/Zeist, which usually is the most expensive region, and Amsterdam, which is in between. On top we also plot log(y t ) (scaled to limit the size of the vertical axis in the graph). Comparing the graphs in Figure 1 suggests that real house prices increase slightly faster than real GDP. Prices in Bunnik/Zeist and Amsterdam show substantial variations in the trend growth rate over time, with alternating periods of steep price increases and of stable or falling prices. Especially the 'hump' in the prices around 2000 stands out clearly. This suggests that the trend in the house prices is stochastic rather than deterministic. Furthermore, as the trending behavior of the different price series seems quite similar regional house prices may well be cointegrated.

Unit roots and cointegration

To test whether these visual impressions from Figure 1 can be given more formal statistical support, we perform panel unit root tests on the regional house prices. Two of the most popular tests in the literature are those from [START_REF] Levin | Unit Root Tests in Panel Data: Asymptotic and Finite-Sample Asymptotics[END_REF] [LLC] and [START_REF] Im | Testing for Unit Roots in Heterogenous Panels[END_REF] [IPS], see [START_REF] Breitung | Unit Roots and Cointegration in Panels[END_REF]. These tests have as null hypothesis the presence of a unit root in all the series in the panel. The alternative hypotheses are different however. [START_REF] Levin | Unit Root Tests in Panel Data: Asymptotic and Finite-Sample Asymptotics[END_REF] assume that the house price dynamics are the same for each region, and therefore the alternative hypothesis is that all regional house prices are stationary. [START_REF] Im | Testing for Unit Roots in Heterogenous Panels[END_REF], however, have as alternative hypothesis that at least F o r P e e r R e v i e w one regional house price is stationary. Both these tests assume that there is no crosscorrelation between different series in the panel. In fact, they are not consistent if such a dependency is present, which is quite likely in our case. Alternative tests that do allow for cross-section dependence are available, like the one in [START_REF] Moon | Testing for a Unit Root in Panels with Dynamic Factors[END_REF], but these usually rely on asymptotics that require T to be much larger then N , while in our case they are about equal.

To meet our data characteristics, we therefore employ the cross-sectionally augmented IPS [CIPS] test, recently developed in [START_REF] Pesaran | A Simple Panel Unit Root Test in the Presence of Cross-Section Dependence[END_REF]. This allows for cross-sectional dependence, and is also valid when N is larger than T . The idea of the CIPS test is to add the cross-section averages of the lagged levels and first differences to the familiar augmented Dickey-Fuller [ADF] regression equation. If it can be assumed that the crosscorrelations are caused by a common factor, then this common factor must also be present in the cross-section averages. Adding these to the ADF equations should then get rid of the common factor in the residuals and thus correct for the presence of cross-correlations.

As the CIPS test is known to have reduced power relative to the IPS and LLC tests in case cross-correlation is not present, we test whether we really should use the CIPS test instead of these simpler tests. For this purpose we use the cross-section dependence

[CD] test of [START_REF] Pesaran | General Diagnostic Tests for Cross Section Dependence in Panels[END_REF] and the adjusted LM [LM adj ] test of [START_REF] Pesaran | A Bias-Adjusted LM Test of Error Cross Section Independence[END_REF].

These tests both use the cross-correlations between the residuals of the individual ADF regressions for the different regions. The CD test takes a simple sum which is scaled such that it has a standard normal distribution under the null hypothesis of no cross-sectional dependence. Therefore, the CD test has little power in case there are both positive and negative correlations such that the average is close to zero. The LM adj test, however, is also valid in this case as it employs the squares of the cross correlations in the construction of the test statistic. However, the LM adj test is less robust against non-normally distributed error terms and exhibits size distortions, especially when N is much larger than T . [Table 1 about here.] Table 1 gives the result of these tests for the panel of quarterly growth rates in house prices ∆ log(p i,t ), where ∆ denotes the first-difference filter, and of log(p i,t ) -log(p 34,t ), which is the difference of each series with the log house prices in Amsterdam (region 34, see Appendix A). The number of lagged (first) differences is allowed to vary across each (C)ADF equation and is determined by minimizing BIC. Adding a lagged variable means losing one observation, therefore we actually minimize BIC/T , see Cameron and Trivedi (2005, pp. 279) or the definition of BIC given in [START_REF] Franses | Quantitative Models in Marketing Research[END_REF]. Each (C)ADF regression equation contains an intercept and a trend.

From the second column of Table 1 we see that for the first difference of the log house prices there is substantial cross-sectional dependence, according to both the CD and LM adj tests. Next, we see that all three unit root tests reject the presence of a unit root in these growth rate series. Results for the difference between the log price in a region and the log price in Amsterdam (region 34) appear in the third column of Table 1. The reason for examining the log price differences with respect to Amsterdam is that finding these to be stationary, we can conclude that the house prices in each region are cointegrated. Again, the CD and LM adj tests indicate that there is substantial cross-sectional dependence.

Next, the LLC and IPS unit root tests do not reject the presence of a unit root, but the CIPS test does. Since the LLC and IPS tests are not valid in case of cross-sectional dependence, we rely on the CIPS test and conclude that the log house prices in each region are cointegrated. Note that the (1, -1) cointegration relationships suggested by the results in Table 1 are quite plausible. It means that the difference between the log of house prices, or, equivalently the ratio of house prices, in each region is a stationary process. This constrains the long-term growth of house prices in each region to be about the same.

[Figure 2 Although the graphs in these figures are rather different, they basically lead to the same conclusion that there are no apparent clusters. Hence, dividing the regions into different groups based only on the cross-correlations of the regional house prices is not a meaningful possibility. Apparently, we need a more sophisticated clustering method, perhaps based on latent classes, as we will propose in the next section.

The model

In this section we put forward the specification of the latent-class panel time series model for describing the regional house prices. We first discuss the characteristics of the model, and then we outline the parameter estimation procedure.

Representation

Our starting point is the latent-class panel time series model developed by [START_REF] Paap | Does Africa Grow Slower Than Asia, Latin America and the Middle East? Evidence from a New Data-Based Classification Method[END_REF]. The crucial idea behind this model is that the individual time series may be grouped into a limited number of clusters. Within each cluster, a linear model is assumed to describe the dynamic behavior of the time series. The clusters are defined such that the model parameters are the same for all time series within a cluster, but they are different across clusters. Hence, this model covers the middle ground between a pooled regression model, where the model parameters are constrained to be the same for all regions, and In our model for quarterly growth rates of house prices we allow for more flexibility than was done in [START_REF] Paap | Does Africa Grow Slower Than Asia, Latin America and the Middle East? Evidence from a New Data-Based Classification Method[END_REF]. As mentioned, there are two research questions we want to answer with our model and each question corresponds to different parameters that can vary across the latent classes. The first is whether the mean growth rates of house prices are the same across all regions. We therefore allow the clusters to have a different average growth rate by allowing for a class-specific intercept. To facilitate interpretation, we demean all other variables in the model such that the intercept is equal to the average growth rate of the house prices in the regions in a cluster.

The second question we wish to answer with our model is whether the house prices in regions follow the trend in real GDP. We add an error correction variable linking regional real house prices and real GDP, where the long-run parameter should be estimated. This long-run parameter determines the size of the effect of GDP on the house prices. The adjustment parameter indicates how fast the house prices in a region react to changes in GDP.

Based on the above discussion, we propose the following latent-class panel time series model for regional house prices in the Netherlands

∆ log(p i,t ) = β 0,k i + β 1,k i [log(p i,t-1 ) + γ k i log(y t-1 )] + η i,t .
(1)

F o r P e e r R e v i e

w

The β and γ parameters are class-specific parameters, where the subscript k i = 1, . . . , K denotes the latent class which region i belongs to with K being the number of latent classes. We denote the probability that a region belongs to latent class k, or the mixing proportions, as π k . Naturally it must hold that, 0 < π k < 1 and that K k=1 π k = 1. As the house prices of each regions are cointegrated with GDP, they are also cointegrated amongst themselves. This can easily be seen in the following way. Both p i,t -γ k i y t and p j,t -γ k j y t are stationary series. Now, consider the following expression,

(p i,t -γ k i y t ) -δ(p j,t -γ k j y t ) = (p i,t -δp j,t ) -(γ k i -δγ k j )y t .
(2)

The LHS of ( 2) is stationary, therefore the RHS is also a stationary series. For δ = γ k i /γ k j the second term on the RHS of (2) will disappear, therefore regions i and j must have a

(1, -δ) cointegration relationship. Two regions in the same cluster will therefore have a

(1, -1) cointegration relationship, because they share the same γ parameter. As we have seen in Section 2.1, there is support for exactly this relationship.

Even though model (1) includes log(y t-1 ), which is the same for all regions, there may still be some cross-section correlation among the house prices that is not captured.

Therefore, following [START_REF] Holly | A Spatio-Temporal Model of House Prices in the US[END_REF], we allow the error term η i,t in (1) to be correlated across regions, but assume that this correlation is due to dependence on certain common factors. To be precise, we consider the specification

η i,t = α 1,i ∆ log(y t-1 ) + α 2,i I t-1 + α 3,i ∆ log(p t-1 ) + ε i,t , (3)
where I t-1 denotes the interest rate at time t -1, p t-1 denotes the average house price in the Netherlands at time t -1 and where α l,i for l = 1, 2, 3 are region-specific parameters.

The residuals ε i,t are now assumed to be independently normally distributed with a regionspecific variance σ 2 i . In the application below, we demean all variables in (1) and ( 3) and hence the inter-

Estimation

The parameters in our model ( 1) with (3) can be estimated as outlined in [START_REF] Paap | Does Africa Grow Slower Than Asia, Latin America and the Middle East? Evidence from a New Data-Based Classification Method[END_REF], using the EM algorithm of [START_REF] Dempster | Maximum Likelihood Estimation from Incomplete Data via the EM Algorithm[END_REF]. This makes use of the full data log-likelihood function, that is, the joint density of the house prices and the latent classes k i , which we specify in detail below. The EM algorithm is an iterative maximization algorithm, which alternates between two steps until convergence occurs. In the first step (E-step) we compute the expected value of the full data log-likelihood function with respect to the latent classes k i , i = 1, . . . , N , given the house prices and the current values of the model parameters. In the second step (M-step) we maximize the expected value of the full data log-likelihood function with respect to the model parameters. As the model given the class memberships can be written as a standard linear regression, the M-step amounts to a series of (weighted) regressions. As the EM algorithm maximizes the log-likelihood function, the resulting estimates of the model parameters are equal to the maximum likelihood [ML] estimates. We can therefore compute standard errors of the estimates using the second derivative of the log-likelihood function.

Note that due to the presence of the term β 1,k i [log(p i,t-1 ) + γ k i log(y t-1 )] the model in ( 1) is actually nonlinear in the parameters. To deal with this issue, we follow [START_REF] Boswijk | Testing for an Unstable Root in Conditional and Structural Error Correction Models[END_REF] and rewrite the model as

∆ log(p i,t ) = β 0,k i + β 1,k i log(p i,t-1 ) + β 2,k i log(y t-1 ) + η i,t , (4)
where The full data likelihood function, that is, the joint density of P = {{∆ log p i,t } T t=1 } N i=1 and K = {k i } N i=1 is given by

β 2,k i = β 1,k i γ k i . Note
l(P, K; θ) = N i=1 K k=1 π k T t=1 1 σ i φ(ε k i,t /σ i ) I[k i =k] , ( 5
)
where φ(•) denotes the probability density function of a standard normal random variable and θ is a vector containing all model parameters. The error term at time t for region i belonging to cluster k is defined as

ε k i,t = ∆ log p i,t -x i,t β k -w t α i , (6)
where x i,t is the (3 × 1) vector with the regressors appearing in (4) and β k contains the corresponding parameters for cluster k. Similarly, w t is the (3 × 1) vector with common factors in the specification for η i,t in (3), and α i = (α 1,i , α 2,i , α 3,i ) containing the parameters for region i.

The expectation of the full data log-likelihood function with respect to K|P, θ [E-step] is given by

L(P; θ) = N i=1 K k=1 πi,k ln π k + T t=1 - 1 2 ln σ 2 i - 1 2 ln 2π - (ε k i,t ) 2 2σ 2 i , ( 7
)
where πi,k denotes the conditional probability that region i belongs to class k. This is

equal to πi,k = π k T t=1 1 σ i φ ε k i,t /σ i K l=1 π l T t=1 1 σ i φ ε l i,t /σ i . ( 8
)
In the M-step, we need to maximize (7) with respect to the parameters β k , π k , k = 1, . . . , K and α i , σ 2 i for i = 1, . . . , N . We perform this maximization step sequentially. First, we optimize over β k keeping the other parameters fixed. This can be done by a simple weighted regression of ∆ log(p i,t ) -w t α i on x i,t with weights given by πi,k /σ i . Second, we optimize the log-likelihood function over α i for i = 1, . . . , N . We do this by regressing K k=1 πi,k [∆ log(p i,t ) -x i,t β k ] on w t . The dependent variable in this regression is the conditional expectation of η i,t . We perform these regressions for each region separately.

Next, the new estimate of σ 2 i is given by

σ 2 i = 1 T T t=1 K k=1 πi,k ε k i,t 2 (9)
for i = 1, . . . , N . Finally, the mixing proportions are updated by averaging the conditional class membership probabilities, that is,

π k = 1 N N i=1 πi,k (10)
for k = 1, . . . , K.

As we maximize over the parameters sequentially in the M-step, we do not reach the optimum of the expected full data log-likelihood function ( 7) in each iteration of the EMalgorithm. We can repeat the individual update steps until convergence, but this is not necessary. Indeed, [START_REF] Meng | Maximum Likelihood Estimation via the ECM Algorithm: a General Framework[END_REF] have shown that an increase in the full-data log-likelihood function in the M-step is sufficient for the EM algorithm to converge to the maximum of the log-likelihood function.

Determining the appropriate number of latent classes is not straightforward. We cannot use a standard statistical test, due to the [START_REF] Davies | Hypothesis Testing when a Nuisance Parameter is Present Only Under the Alternative[END_REF] problem of unidentified . [START_REF] Bozdogan | Mixture-Model Cluster Analysis Using Model Selection Criteria and a New Information Measure of Complexity[END_REF] suggests that the AIC should have a penalty factor of 3 instead of 2 in the case of mixture models. Indeed, [START_REF] Andrews | A Comparison of Segment Retention Criteria for Finite Mixture Logit Models[END_REF] show that this AIC-3 criterion outperforms other criteria. [START_REF] Bozdogan | Model Selection and Akaike's Information Criterion (AIC): The General Theory and its Analytical Extensions[END_REF] modifies the AIC into the so-called consistent Akaike information criterion [CAIC], which is almost equal to BIC. He shows that when the sample size is large the CAIC and BIC criteria perform better than AIC. We will consider all four criteria below.

Empirical results

In this section we discuss the results of applying our model to the regional house price data for the Netherlands described in Section 2. The effective sample period ranges from

A fully heterogenous model

We first estimate the parameters in a fully heterogenous model, that is, we estimate the model in ( 1 [Table 2 about here.]

[Figure 4 about here.]

The mean, minimum and maximum of each parameter of the 76 regions can be found in Table 2. Figure 4 displays the histograms for the 76 estimated values for each of the parameters β j , j = 0, 1, and γ in (1). The top panel shows the intercepts, β 0 , which equal the quarterly growth rates. These are all positive, reflecting the upward trend in the house prices, and range between 0.6% and 1.3% per quarter. The middle panel of Figure 4 shows the results for the adjustment parameter for the cointegration term with GDP. We find some positive values, which is not as expected, as these imply divergence between GDP and the house prices in that region. Finally, the histogram in the bottom panel shows the parameter γ in the cointegration relationship with GDP, which we expect to be negative as we expect the house prices and GDP to move in the same direction.

Table 2 also shows the results for the α parameters from (3). Again we find that they show some counterintuitive signs and a relatively large spread.

We can see from these results that some form of aggregation may be useful, as we now get a wide variety of parameter estimates, with sometimes quite implausible results. At the same time, this variety also suggests that we should perhaps better not restrict the parameters to be the same across all regions. Hence, it may be optimal to allow for a limited number of different clusters.

A model with latent classes

[Table 3 we consider four different information criteria for this purpose. Table 3 shows the values of these criteria for models with one to four and 76 classes. For all criteria, we see that going from a homogenous model (with a single class) to two classes amounts to a relatively large improvement in the balance of model fit and complexity. After this, adding more classes does not improve any of the criteria. We therefore focus on the model with two latent classes.

[Table 4 about here.]

The estimation results for the model with two latent classes are given in Table 4.

Additionally, Table 5 gives the results for a series of Wald tests which we use to examine whether the parameters for the different classes are significantly different from each other.

The estimation results show that the regions in the two latent classes do indeed differ from each other in several important respects. First, the estimated intercepts show that the average growth rate in class 1 is slightly higher than in class 21 . This difference is not significant though, as can be seen from the second row of Table 5. The average growth rate in class 1 is equal to 1.2% per quarter, or 4.8% annually, while the house prices in class 2 grow with 1.1% per quarter, or 4.4% annually.

Second, examining the cointegration relationship with GDP, we find that class 1 has a significantly larger adjustment parameter. Thus, the house prices in regions belonging to cluster 1 react faster to changes in GDP than the house prices in class 2.

[Table 5 long run the house prices in the regions in this cluster grow almost twice as fast as GDP.

In class 2 the cointegration relationship is (1, -1.68). These long term relationships may not be very plausible, however, as we could already see from Figure 1, they are a good description of the development of house prices and GDP in the sample period.

[Table 6 about here.]

As we showed in Section 3, the cointegration relationship of each region with GDP entails that the regions are also cointegrated among themselves. The long term parameter is only influenced by the γ parameters of the two regions involved, and thus only depends on the class membership of the two regions. Table 6 shows these cointegration relationships between the house prices of regions from any of the two clusters. First, we see that two regions that belong to the same cluster are (1, -1) cointegrated. This is actually very intuitive, as they have follow the same trend relative to the trend of GDP, they must follow the exact same trend themselves. Next we find that a region from cluster 1 is

(1, -1.12) cointegrated with a region from cluster 2. This corresponds with the slightly higher growth rate in class 1.

The parameters in (3) are region-specific, and full estimation results are not reported to save space. Only 11% of the α 1,i parameters is significant, suggesting that the impact of GDP on the house prices is mostly captured by the cointegration term. Moreover, only 22% has the expected positive sign. The α 2,i parameters are mostly negative, and only one region has an (insignificant) positive value. Furthermore, for 63% of the regions the α 2,i

parameter is significant at the 5% level, indicating that the interest rate indeed influences the house prices in the expected direction. The α 3,i parameters, relating the growth of the house price in a region to growth of the average house price in the Netherlands in the previous quarter, is positive for 88% of the regions, but only significant for 42% of these regions. The parameter estimation results obviously become more interesting if we know which regions belong to each of the two classes. Therefore, we compute the conditional class membership probabilities using (8). The resulting classification of the regions is shown in Figure 5. Regions are colored based on πi,1 , the probability of belonging to class 1.

Regions are colored in four shades of grey. For the regions that are colored in the lightest shade it holds that πi,1 ≤ 0.2. For regions colored in subsequently darker shades of grey it holds that 0.2 < πi,1 ≤ 0.4, 0.4 < πi,1 ≤ 0.6, or 0.6 < πi,1 ≤ 0.8. There were no regions with πi,1 > 0.8. It can be seen that most regions are either very dark or very light, suggesting that the classification is very clear for most regions. In fact, the average value of max(π i,1 , πi,2 ) is equal to 0.83.

We find that class 1 contains mainly rural regions surrounding the big cities in the regions in class 1. First, the average income in these regions is likely to increase, as the individuals who move away from the cities are relatively wealthy. The second consequence is an increase of housing quality in these regions, as wealthier people leaving the cities will increase the demand for more luxurious houses.

These potential structural changes within the regions of cluster 1 are consistent with all of our findings. First, the increase in housing quality will result in a larger increase in the average house prices in class 1 as compared to class 2. Our second finding is that house prices in these regions react faster to changes in GDP. This may be caused by tha fact that the increase of their income may influence the decision of these individuals to move and start commuting. Our last and most striking finding is that the house prices in class 1 increase almost twice as fast as GDP. Note however that the increase is not corrected for higher housing quality.

Impulse-response functions

To give further interpretation to our estimation results we compute impulse-response functions for two interesting scenarios, each occurring in the second quarter of 2005. In the first scenario real GDP receives a shock of 1%. In the second scenario real GDP stays the same, but the interest rate receives a shock of 1%-point. We forecast the house prices for each of the scenarios and compare with a no-change scenario, for the subsequent three-year period from 2005Q3 until 2008Q2.

In order to compute the impulse responses up to 12 quarters ahead, we also need forecasts for GDP and the interest rate, as these variables also affect house prices, see

(1). Here we assume that the interest rate stays the same during the forecast period. In scenario 3, the interest rate is higher, but still assumed to be constant over the whole forecast period. To obtain forecasts for GDP we construct a simple AR(q) model with intercept for ∆ log y t . We choose q based on out-of-sample forecasting performance, where we use the last 3 years as a hold-out sample. It turns out that q = 8 gives the best performance.

[Figure 6 about here.]

Figure 6 shows the impulse-response functions of the log house prices with respect to the log of GDP. The y-axis gives the relative change in house prices between the two scenarios, that is, a value of 0.01 means that the house price is 1% higher than the reference forecast. We calculate the impulse response functions for each of the 76 regions.

We then aggregate these to average responses in the two clusters.

We find that the effect of an increase in GDP is initially negative in both clusters, which is caused by the many negative α 1,i parameter in both clusters. However, this negative effect lasts only one quarter, and after that the house prices are higher compared to the reference forecasts. As expected, we find that the house prices in cluster 1 react both faster and more on the change in GDP.

[Figure 7 about here.]

In the second scenario, the interest rate receives a shock, and increases from 2.06% to 3.06%. We find that the house prices are falling. After three years the house prices are about 2% lower in lower in cluster 2 and almost 3% lower in cluster 1, as compared to the reference forecasts.

Conclusions

In this paper we developed a latent-class panel time series model for describing several key characteristics of regional house prices in the Netherlands between 1985 and 2005. An important feature of the model is that we cluster the regions in separate classes, where the price dynamics of house prices in regions within the same class are similar, while they The second class contains both the larger cities and some more remote rural regions.

The house prices in regions in the first class are characterized by slightly higher average growth rates, and stronger and faster reactions to changes in GDP. These findings may be caused by the increased number of commuters. Indeed, the number of people working in the larger cities, but living in the regions of class 1, has increased substantially during our sample period.

Our model allows for the analysis of rather detailed data. 33

  about here.] to our conditional clustering analysis using latent class techniques we consider unconditional clustering based on the correlations of the house price growth rates or of the residuals of the ADF regressions used above. For this purpose, we use multi-dimensional scaling[MDS], which results in the graphs shown in Figure2and 3.

  heterogenous' model, where the parameters are allowed to be different for each individual region. Whereas a pooled regression model may be too restrictive, a fully heterogenous model may be too flexible and ignores the possible similarities between regions. Finally, the key feature of the model of[START_REF] Paap | Does Africa Grow Slower Than Asia, Latin America and the Middle East? Evidence from a New Data-Based Classification Method[END_REF] is that the number of clusters in the model as well as the allocation of the individual time series to different clusters is purely data-based. This avoids ex ante, and necessarily subjective, grouping of regions according to geographical location or economic or demographic characteristics, for example.

  β 0,k i in (1) are equal to the average growth rates of the house prices in the latent classes k i for k i = 1, . . . , K.

  want regions with a larger probability of belonging to class k to have a larger weight in estimating β k . At the same time, regions with a larger standard deviation of the error term σ i should get a smaller weight, as their house prices contain relatively more noise and less information about β k . Each β k , k = 1, . . . , K is estimated in a separate weighted regression.

  the null hypothesis. The usual approach is using a criterion function balancing the fit and the complexity of the model, where the model fit is measured by the value of the log-likelihood function while the number of model parameters provides a measure of complexity. The most well-known criteria are the Akaike information criterion [AIC] and the Bayesian information criterion[BIC]

  1985Q3 (because we have ∆ log(p t-1 ) = log(p t-1 ) -log(p t-2 ) in our model) to 2005Q2 (because we only have real GDP data until 2005Q2), giving T = 80 data points in the time series dimension. To obtain a first impression of the extent of similarities across regions, we start by estimating a fully heterogenous model allowing for different parameters for each region. Next, we provide estimation results for the model with a limited number of latent classes. Finally, we consider impulse-response functions for three interesting scenarios to provide further interpretation of the model.

  ) with (3) allowing for different parameters for each individual region. This 13 model with K=76 latent classes, in which case each region forms a separate class.

  Netherlands. The regions in this class mainly cover parts of Noord-Brabant and the Veluwe. Even though the East belongs almost completely to class 1, the larger cities of the East, like Zwolle, Almelo, Hengelo, Enschede, and Arnhem are part of class 2. Class 2 contains different types of regions. First, it contains many large cities in different parts of the country, like Breda and Groningen, as well as almost all of the regions in the Randstad, the densely populated western part of the country. At the same time some rural regions, like Zeeland, Zuid-Limburg and regions in the North belong to this class with high probability. Note that these rural regions are not as close to the Randstad as most of those in class 1.A possible explanation for our results is the increased number of commuters that live in the regions belonging to class 1 and who work in the large western cities. If the number of commuters increases, it is likely that they move to regions in cluster 1, as these are still at traveling distance from the Randstad. This development has two consequences for the

  the classes. For the 76 regions in the Netherlands we find that two classes are sufficient. The first class contains mainly rural regions close to large cities.

Figure 1 :Figure 2 :Figure 3 :Figure 4 :Figure 5 :Figure 6 :Figure 7 :

 1234567 Figure 1: Log house prices for 3 distinct regions, and log GDP.

  model US house prices at the state level. Their model is 'fully heterogenous' in the sense that it has different parameters for each region.

  that (4) is linear in the parameters, which facilitates estimation. The ML estimate γk i can be obtained from the ML estimates of β 1,k i and β 2,k i as β2,k i / β1,k i .

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  A major issue for successful application of the latent-class panel time series model is of course determining the appropriate number of latent classes. As discussed in Section 3.2,

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  To fully exploit its properties one would want to analyze even further disaggregated data. The collection of such more detailed series is left to further research. Another issue for further research is to make the class probabilities dependent on certain explanatory variables.

		Submitted Manuscript
	A Regions by number
	1 Noordoost-Groningen 27 Kop v. Noord-Holland 52 Dordrecht
	2 Slochteren +s	28 Noord-Kennemerland	53 Gorinchem
	3 Grootegast +s	29 West-Friesland		54 Culemborg/Dodewaard
	4 Stad Groningen +s	30 Midden-Kennemerland 55 Ede +s
	5 Zuidoost-Groningen	31 Waterland		56 Arnhem
	6 Noord-Drenthe	32 Zaanstreek		57 Duiven/Westervoort
	7 Opsterland	33 Zuid-Kennemerland	58 Elst +s
	8 Oost-Friesland	34 Amsterdam		59 Nijmegen
	9 Noordwest-Friesland 35 De Bollenstreek	60 Noordoost-Brabant
	F o r F o r 10 Zuidwest-Friesland 36 Haarlemmermeer 11 Zuid-Friesland 37 Almere F 12 Zuidwest-Drenthe 38 Het Gooi 13 Zuidoost-Drenthe 39 Amersfoort o 14 Hardenberg +s 40 Barneveld r 15 Kop van Overijssel 41 Bunnik/Zeist	61 Uden +s 62 Oss +s 63 Den Bosch 64 Waalwijk/Drunen 65 Zeeuwse Eilanden 66 Zeeuws-Vlaanderen
	16 Zwolle +s 17 Raalte +s 18 Almelo Tubbergen 19 Hengelo Enschede 20 Ruurlo Eibergen 21 Doetinchem +s	P e e r P e e r 42 Utrecht P 43 Woerden 44 Alphen e 45 Leiden e 46 Den Haag r 47 Gouda		67 Bergen op Zoom +s 68 West-Brabant 69 Breda 70 Tilburg/Oirschot 71 Eindhoven +s 72 Zuidoost-Brabant
	22 Zutphen +s 23 Apeldoorn +s 24 Nunspeet +s 25 Lelystad	R e R e 48 Delft +s 49 Rotterdam R 50 Westland e 51 Brielle/Goeree	73 Noord-Limburg 74 Weert +s 75 Roermond +s 76 Zuid-Limburg
	26 Den Helder/Texel Note: +s means including surrounding area.	v i e v i e v e i
				w w w

Table 1 :

 1 Results of the CD test, the LM adj test and three different tests for a unit root for two series (boldface numbers indicate rejection of the null hypothesis). Test statistic is asymptotically distributed as normal b Tables with critical values for various values for Nand T are given by[START_REF] Pesaran | A Simple Panel Unit Root Test in the Presence of Cross-Section Dependence[END_REF], in the presence of an intercept and a trend in the CADF equations and for N = T = 70 the critical value at the 95%level is -2.58, for N = T = 100 it is -2.56.

			Submitted Manuscript
	Test Series ∆[log(p i,t )] log(p i,t ) -log(p 34,t ) CD a 92.0 144.2
	LM adj	a	60.4	175.1
	LLC a		-61.2	2.0
	IPS a		-55.9	1.9
	CIPS b	-8.9	-3.5

a 31 Page 33 of 37 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Table 2 :

 2 Results for the fully heterogenous model.

	Parameter Mean Minimum Maximum
	β 0	0.011	0.006	0.013
	β 1	-0.363	-0.692	0.125
	γ	-0.591	-7.601	2.127
	α 1	-0.271	-1.851	0.703
	α 2	-0.004	-0.013	0.004
	α 3	0.277	-0.340	0.739

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Table 3 :

 3 Criteria values for different numbers of latent classes (boldface numbers indicate the optimum).

	Criterion \ K	1	2	3	4	76
	AIC

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Table 4 :

 4 Estimation results for K = 2 latent classes.

	Class Estimate Standard error
		intercept β 0,k
	1	0.012	0.001
	2	0.011	0.000
	adjustment parameter GDP β 1,k
	1	-0.178	0.019
	2	-0.131	0.007
	cointegration relationship GDP γ k
	1	-1.888	0.083
	2	-1.684	0.035
		mixing proportions π k
	1	0.202	0.159
	2	0.798	.

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Table 5 :

 5 Wald tests for equality of the parameters across the two classes in (1).

	Restriction Wald statistic p-value
	β 0,1 = β 0,2	0.61	0.41
	β 1,1 = β 1,2	7.10	0.01
	γ 1 = γ 2	6.90	0.01

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Table 6 :

 6 Cointegration relationships between the regions from clusters i and j.

	i \ j	1	2
	1	(1, -1)	(1, -1.12)
	2	(1, -0.89)	(1, -1))

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

Recall that we demeaned all other variables the model, so the intercepts represent the average growth rates.

Page 17 of 37 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript