Session 2: Fixed and Random Effects Estimation

Transcription

1 Session 2: Fixed and Random Effects Estimation Principal, Developing Trade Consultants Ltd. ARTNeT/RIS Capacity Building Workshop on the Use of Gravity Modeling Thursday, November 10,

2 Outline Fixed Effects Estimation 1 Fixed Effects Estimation

3 Using Dummies to Capture Multilateral Resistance The AvW Gravity Model Aggregate Data ) ] X ij = ln(y i ) + ln (E j ln(y ) + (1 σ) [lnτ ij + lnπ i + lnp j The term ln(y ) is common across all exporters and importers; thus, it can be captured through a constant in the regression model. ) The term ln (E j + lnp j is constant across all importers for a given exporter; thus, it can be captured through an importer dummy variable (fixed effect). The term ln ( ) Y i + lnπi is constant across all exporters for a given importer; thus, it can be captured through an 3 exporter dummy variable (fixed effect).

4 Using Dummies to Capture Multilateral Resistance Advantages of Dummy Variables An aggregate gravity model with a constant, and dummy variables for each exporter and each importer will therefore take proper account of multilateral resistance, and should produce unbiased estimates. Very simple to estimate, but takes account of some sophisticated effects. NxN observations, but N+N dummies; degrees of freedom are usually sufficient. 4

5 Using Dummies to Capture Multilateral Resistance Disadvantages of Dummy Variables Dimensionality quickly becomes an issue with sectoral models: N+N can be in the hundreds, or thousands. Because of collinearity constraints, we cannot identify separate effects due to factors that vary in the exporter or importer dimensions. Only factors varying bilaterally can be identified. 5

6 Using Dummies to Capture Multilateral Resistance Estimation using panel data techniques can make it possible to reduce the dimensionality problem somewhat, but it remains an issue in large/detailed datasets. To deal with the collinearity problem, variables can sometimes be transformed so as to vary by country pair: Sum of exporter and importer values. Average of exporter and importer values, etc. Try to go back to theory to see if this is an appropriate thing to do in a given circumstance. 6

7 Dummies in Sectoral Gravity Models As suggested previously, things get even more complicated with sectoral gravity models. Dummy variables need to be specified in the importer-sector, exporter-sector, and sector dimensions, because: ( ln X k ij ) ( = ln Yi k (1 σ k ) ) ( + ln E k j ) ( ln [ lnτ k ij + lnπ k i + lnp k j Y k) + ] In addition, trade costs need to be interacted with sector dummies in order to take account of varying elasticities of substitution across sectors. 7

8 Dummies in Sectoral Gravity Models Depending on the level of sectoral disaggregation used, this approach can result in huge numbers of parameters. Models can take a long time, and a big computer, to estimate. It is usually much easier to estimate separate models for each sector. 8

9 Dummies and Fixed Effects in Stata Option 1: enter the dummies manually and use OLS: tab importer, gen(imp_dum_*) reg ln_trade... imp_dum_*, robust Option 2: use a panel estimator (OLS + a trick) to account for one set of dummies: iis importers xtreg ln_trade..., robust fe 9

10 Random Effects: An Alternative to Dummies Fixed effects (dummy variables) are one way of accounting for unobserved heterogeneity across countries, in this case due to multilateral resistance. A common alternative in the econometrics literature is random effects: Fixed effects allow for free or structureless variation; Random effects require that unobserved heterogeneity obey some probability constraints, i.e. follow a particular distribution. 10

11 Random Effects: An Alternative to Dummies Advantages of Random Effects The dimensionality constraint is greatly relaxed: even very large models can be estimated relatively quickly. Allows inclusion of variables (like GDP) that vary in the same dimension as the random effects. Simple to estimate single-dimensional RE models in Stata: iis importers xtreg ln_trade ln_gdp_imp [etc.], re robust. 11

12 Random Effects: An Alternative to Dummies Disadvantages of Random Effects Random effects rely on a strong assumption: multilateral resistance is normally distributed across countries, with a given standard deviation. The AvW model tells us that multilateral resistance is important, but it doesn t tell us anything about its distribution. In practice, compare RE and FE estimates. 12

13 Random Effects: An Alternative to Dummies Distance coefficient from the original gravity model without fixed effects = ***. Distance coefficient from the fixed effects gravity model = ***. Distance coefficient from the random effects (importer) gravity model = *** Not statistically different from OLS, but significantly different from fixed effects. CAUTION! 13

14 Testing for Fixed vs. Random Effects Fixed effects estimates are always consistent, even if the true model is random effects. Random effects are only consistent if the true model is random effects, in which case they are also efficient. Intuitively, random effects are an acceptable simplification when the difference between the two sets of coefficients is small. 14

15 Testing for Fixed vs. Random Effects The Hausman test formalizes this intuition: it tests for a statistically significant difference between the two sets of coefficients. If the null hypothesis is rejected, random effects are inappropriate (inconsistent). If the null hypothesis is not rejected, random effects are an acceptable simplification, but either estimator can be used. 15

16 Testing for Fixed vs. Random Effects Beware: the Hausman test has poor properties in practice! Stata s implementation has to use non-robust covariance matrices. Sometimes, the test statistic cannot even be calculated due to a breakdown in its assumptions. If you are serious about random effects, try the Hausman test, but take it with a grain of salt. 16

17 Testing for Fixed vs. Random Effects In Stata, adopt the following procedure to test for fixed versus random effects: Estimate the fixed effects model without the robust option, then issue the command estimates store fixed. Estimate the random effects model without the robust option, then issue the command estimates store random. Issue the command hausman fixed random. 17

18 The Baier-Bergstrand Model It would be nice to have a methodology that enables us to account for the MR terms, but also include data that only varies by exporter or importer. Random effects can do the job, but only at the price of a strong assumption. Fixed effects requires us to play with the data to make variables that vary (artificially) by country pair. Baier and Bergstrand (2009 JIE) give us an alternative. 18

19 The Baier-Bergstrand Model Recall that the MR terms are complex, non-linear functions of trade costs in the AvW model: } ( ) 1 σk Π k 1 σk i = C Ej k j=1 ( P k j { τ k ij P k j ) 1 σk = C j=1 { τ k ij Π k i Y k } 1 σk Y k i Y k The Baier-Bergstrand model uses a first order Taylor series expansion to approximate the MR terms. 19

20 The Baier-Bergstrand Model The Taylor series approach gives a gravity model that looks like this: ln ( ) X ij = ln(yi ) + ln ( ) E j ln(y )+ [ (1 σ) lnτ ij i θ i lnτ ij j θ j lnτ ij + i ] θ i θ j lnτ ij j The two θ terms are GDP weights, i.e. θ i = Y i Y. 20

21 The Baier-Bergstrand Model To estimate the Baier-Bergstrand model, simply: Calculate the GDP weights. For each trade cost variable lnτ ij calculate lnτij = lnτ ij i θ i lnτ ij j θ j lnτ ij + i j θ i θ j lnτ ij. Remember to take logs before taking the averages! Estimate the gravity model using OLS, as usual, but including the transformed variables along with GDP, i.e.: ln ( X ij ) = ln(yi ) + ln ( E j ) ln(y ) + lnτ ij 21

22 The most common approach to the aggregate gravity model is to use fixed effects (dummy variables) by importer and by exporter. Random effects can also be used, but they rely on a stronger and possibly invalid assumption. For sectoral gravity models, the simplest approach is to estimate separately, sector by sector. The Baier-Bergstrand model provides a simple alternative that makes it possible to account for multilateral resistance at the same time as including exporter- and importer-specific variables. 22