Chapter 4: Test Hypotheses with Panel Data

Chapter 4: Test Hypotheses with Panel Data
This chapter introduces statistical tests of important hypotheses for the panel data models. Main tests include: Tests for poolability Tests for fixed effects -- F-tests Tests for random effects LR tests for random effects Wald tests for random effects LM tests for random effects Hausman’s tests of Random Effects vs Fixed Effects Hausman’s test for the one-way model Hausman’s test for the two-way model Zhenlin Yang

4.1. Tests for Poolability Recall the panel data model with two-way effects given in (3.1): 𝑦 𝑖𝑡 =𝛼+ 𝑋 𝑖𝑡 ′ 𝛽+ 𝑢 𝑖𝑡 , 𝑢 𝑖𝑡 = 𝜇 𝑖 + 𝜆 𝑡 + 𝑣 𝑖𝑡 , i = 1, , N and t = 1, , T. This model assumes that (𝛼, 𝛽) are constant over i and t. Question of whether or not to pool data arises naturally when one is dealing with panel data. This boils down to tests whether (𝛼, 𝛽) vary over i, i.e., whether the model 𝑦 𝑖𝑡 = 𝛼 𝑖 + 𝑋 𝑖𝑡 ′ 𝛽 𝑖 + 𝑢 𝑖𝑡 is more appropriate; or, to tests whether (𝛼, 𝛽) vary over t, i.e., whether the model 𝑦 𝑖𝑡 = 𝛼 𝑡 + 𝑋 𝑖𝑡 ′ 𝛽 𝑡 + 𝑢 𝑖𝑡 is more appropriate. Baltagi (2013, Sec. 4.1) presents a couple of tests for the first case. Here, we focus on other type of tests, assuming poolability.

𝐻 0 : 𝜇 1 = ⋯= 𝜇 𝑁 =𝛼, given 𝜆 1 =…= 𝜆 𝑇 =𝛼.
4.2. Test for Fixed Effects Consider the two-way fixed effects model given Ch. 3: 𝑦 𝑖𝑡 =𝛼+ 𝑋 𝑖𝑡 ′ 𝛽+ 𝜇 𝑖 + 𝜆 𝑡 + 𝑣 𝑖𝑡 , i = 1, , N and t = 1, , T. First, we are interested in a conditional test of the hypothesis: 𝐻 0 : 𝜇 1 = ⋯= 𝜇 𝑁 =𝛼, given 𝜆 1 =…= 𝜆 𝑇 =𝛼. If 𝑣 𝑖𝑡 are iid N(0, 𝜎 𝑣 2 ), an F test was given in Ch. 2 as follow: 𝐹 C1 = (RRSS−URSS)/(𝑁−1) URSS/(𝑁𝑇−𝑁−𝐾) 𝐻 0 ~ 𝐹 𝑁−1, 𝑁 𝑇−1 −𝐾 where RRSS is the restricted residual sum of squares from fitting the null model with df = NT(K+1); URSS is the unrestricted residual sum of squares from fitting Model (2.2) by LSDV or within method with df = NTNK.

Conditional Tests for Fixed Effects
Similarly, we have another conditional test of the hypothesis: 𝐻 0 : 𝜆 1 = ⋯= 𝜆 𝑇 =𝛼, given 𝜇 1 =…= 𝜇 𝑁 =𝛼. Paralleled with the development of the 𝐹 C1 test, an F test for testing the lack of time FE, given that there are no individual FE is given as follow: 𝐹 C2 = (RRSS−URSS)/(𝑁−1) URSS/(𝑁𝑇−𝑁−𝐾) 𝐻 0 ~ 𝐹 𝑇−1, 𝑁−1 𝑇−𝐾 where RRSS is the restricted residual sum of squares from fitting the null model with df = NT(K+1); URSS is the unrestricted residual sum of squares from fitting a model with time FE only by LSDV or within method with df = NTTK.

Joint Test for Fixed Effects
A test of the joint significance of the individual and time fixed effects, i.e., a joint test of the hypothesis: 𝐻 0 : 𝜇 1 = ⋯= 𝜇 𝑁 =𝛼 and 𝜆 1 = ⋯= 𝜆 𝑇 =𝛼, Is of interest. If 𝑣 𝑖𝑡 are iid N(0, 𝜎 𝑣 2 ), the an F test was given in Ch. 3 as follow: 𝐹 𝐽 = (RRSS−URSS)/(𝑁+𝑇−2) URSS/((N  1)(T 1)−𝐾) ~ 𝐹 𝑁+𝑇−2, (𝑁−1) 𝑇−1 −𝐾 RRSS: the restricted residual sum of squares from fitting the null model, with NT(K+1) degrees of freedom (df). URSS: the unrestricted residual sum of squares from fitting Model (3.2), with df = (N1)(T1)K. Stata does not have commands for two-way FE analyses/tests.

Marginal Tests for Fixed Effects
Chap 3 also introduced a marginal test for the non-existence of individual FE, allowing for the existence of time FE: 𝐻 0 : 𝜇 1 = ⋯= 𝜇 𝑁 =𝛼 allowing 𝜆 𝑡 ≠𝛼, 𝑡=1, , 𝑇. Under the null, one fits the model: 𝑦 𝑖𝑡 =𝛼+ 𝑋 𝑖𝑡 ′ 𝛽+ 𝜆 𝑡 + 𝑣 𝑖𝑡 by LSDV; otherwise, fits Model (3.2) by Within estimator. Under Assumption A and normality of 𝑣 𝑖𝑡 , an F test is 𝐹 M1 = (RRSS−URSS)/(𝑁−1) URSS/((N  1)(T 1)−𝐾) ~ 𝐹 𝑁−1, (𝑁−1) 𝑇−1 −𝐾 RRSS: the restricted residual sum of squares from fitting the null model, with df = NTTK. URSS: the unrestricted residual sum of squares from fitting Model (3.2), with df = (N1)(T1)K.

Marginal Test for Fixed Effects
Paralleled with the above, a marginal test for the non-existence of time FE, allowing for the existence of individual FE: 𝐻 0 : 𝜆 1 = ⋯= 𝜆 𝑇 =𝛼 allowing 𝜇 𝑖 ≠𝛼, 𝑖=1, , 𝑁, is of interest. Under 𝐻 0 , one fits the model: 𝑦 𝑖𝑡 =𝛼+ 𝑋 𝑖𝑡 ′ 𝛽+ 𝜇 𝑖 + 𝑣 𝑖𝑡 by LSDV; otherwise, Model (3.2) by Within estimator. Under Assumption A and normality of 𝑣 𝑖𝑡 , an F test is 𝐹 M2 = (RRSS−URSS)/(𝑁−1) URSS/((N  1)(T 1)−𝐾) ~ 𝐹 𝑇−1, 𝑁−1 (𝑇−1)−𝐾 RRSS: the restricted residual sum of squares from fitting the null model, with df = NTN K. URSS: the unrestricted residual sum of squares from fitting Model (3.2), with df = (N1)(T1) K.

Test for Fixed Effects: Stata FC1 -Tests
Table 4.1. Public Capital Productivity: One-way Individual FE. . xtset state0 . xtreg ln_gsp ln_pcap ln_pc ln_emp unemp, fe Fixed-effects (within) regression Number of obs = Group variable: state Number of groups = R-sq: Obs per group: within = min = between = avg = overall = max = F(4,764) = corr(u_i, Xb) = Prob > F = ln_gsp | Coef. Std. Err t P>|t| [95% Conf. Interval] ln_pcap | ln_pc | ln_emp | unemp | _cons | sigma_u | gives the standard deviation of individual effects 𝜇 𝑖 sigma_e | gives the standard deviation of idiosyncratic error 𝑣 𝑖𝑡 rho | (fraction of variance due to u_i) F test that all u_i=0: F(47, 764) = Prob > F =

Test for Fixed Effects: Stata FC2 -Tests
Table 4.2. Public Capital Productivity: One-way Time FE. . xtset yr . xtreg ln_gsp ln_pcap ln_pc ln_emp unemp, fe Fixed-effects (within) regression Number of obs = Group variable: yr Number of groups = R-sq: Obs per group: within = min = between = avg = overall = max = F(4,795) = corr(u_i, Xb) = Prob > F = ln_gsp | Coef. Std. Err t P>|t| [95% Conf. Interval] ln_pcap | ln_pc | ln_emp | unemp | _cons | sigma_u | sigma_e | rho | (fraction of variance due to u_i) F test that all u_i=0: F(16, 795) = Prob > F =

Test for Fixed Effects: Stata FM1 -Tests
Table 4.3. Public Capital Productivity: Individual FE, Time Dummies. . xtreg ln_gsp ln_pcap ln_pc ln_emp unemp i.yr, fe Fixed-effects (within) regression Number of obs = Group variable: state Number of groups = R-sq: Obs per group: within = min = between = avg = overall = max = F(20,748) = corr(u_i, Xb) = Prob > F = ln_gsp | Coef. Std. Err t P>|t| [95% Conf. Interval] Covariates … Time Dummies … sigma_u | sigma_e | rho | (fraction of variance due to u_i) F test that all u_i=0: F(47, 748) = Prob > F =

Test for Fixed Effects: Stata FM2-Tests
Table 4.4. Public Capital Productivity: Time FE, Individual Dummies. . xtreg ln_gsp ln_pcap ln_pc ln_emp unemp i.state0, fe Fixed-effects (within) regression Number of obs = Group variable: yr Number of groups = R-sq: Obs per group: within = min = between = avg = overall = max = F(51,748) = corr(u_i, Xb) = Prob > F = ln_gsp | Coef. Std. Err t P>|t| [95% Conf. Interval] Covariates … Individual Dummies … sigma_u | sigma_e | rho | (fraction of variance due to u_i) F test that all u_i=0: F(16, 748) = Prob > F =

4.3. Tests for Random Effects
Consider the two-way random effects model given Ch. 3: 𝑦 𝑖𝑡 =𝛼+ 𝑋 𝑖𝑡 ′ 𝛽+ 𝜇 𝑖 + 𝜆 𝑡 + 𝑣 𝑖𝑡 , i = 1, , N and t = 1, , T. Recall Assumption C: (i) 𝜇 𝑖 ~ IID(0, 𝜎 𝜇 2 ), 𝜆 𝑡 ~ IID(0, 𝜎 𝜆 2 ), and 𝑣 𝑖𝑡 ~ IID(0, 𝜎 𝑣 2 ), independent of each other, and (ii) 𝑋 𝑖𝑡 is independent of 𝜇 𝑖 , 𝜆 𝑡 , and 𝑣 𝑖𝑡 for all i and t. A joint test of no individual and time random effects is to test 𝐻 0 : 𝜎 𝜇 2 = 𝜎 𝑣 2 =0. A conditional test for individual RE, given no time RE is to test 𝐻 0 : 𝜎 𝜇 2 =0, given 𝜎 𝑣 2 =0. A conditional test for time RE, given no individual RE is to test 𝐻 0 : 𝜎 𝑣 2 =0, given 𝜎 𝜇 2 =0.

LR Tests for Random Effects
A marginal test for individual RE, allowing time RE is to test 𝐻 0 : 𝜎 𝜇 2 =0, allowing 𝜎 𝑣 2 >0. A marginal test for time RE, allowing individual RE is to test 𝐻 0 : 𝜎 𝑣 2 =0, allowing 𝜎 𝜇 2 >0. A general test for testing the above hypotheses is the likelihood ratio (LR) test, which reads as: “the negative twice the difference between the maximized loglikelihood of the restricted model and that of the unrestricted model”, i.e., 𝐿𝑅=−2( ℓ 𝑅 − ℓ 𝑈 ), where ℓ 𝑅 is the maximum of the loglikelihood of the restricted model, and ℓ 𝑈 is that of the unrestricted model. The limiting null distribution of LR is chi-squared, The df of LR test = no. of restrictions imposed by 𝐻 0 . See Greene (2012, Sec. 14.6) for details.

LR Tests for Random Effects -- Stata
Table 4.5. Public Capital Productivity: One-way Individual RE. . xtset state0 panel variable: state0 (balanced) . xtreg ln_gsp ln_pcap ln_pc ln_emp unemp, mle Fitting constant-only model: … Iteration 2: log likelihood = Fitting full model: Iteration 0: log likelihood = Iteration 4: log likelihood = Random-effects ML regression Number of obs = Group variable: state Number of groups = Random effects u_i ~ Gaussian Obs per group: min = avg = max = LR chi2(4) = Log likelihood = Prob > chi =

Table 4.6. Public Capital Productivity: Two-Way Random Effects . xtset state0 yr panel variable: state0 (strongly balanced) time variable: yr, 1970 to 1986 delta: 1 unit . xtmixed ln_gsp ln_pcap ln_pc ln_emp unemp || _all: R.yr || state0:, mle Performing EM optimization: Performing gradient-based optimization: Iteration 0: log likelihood = Iteration 1: log likelihood = Computing standard errors: Mixed-effects ML regression Number of obs = | No. of Observations per Group Group Variable | Groups Minimum Average Maximum _all | state0 | Wald chi2(4) = Log likelihood = Prob > chi =

Public Capital Productivity Data: Two-Way Random Effects, Cont’d From Table 4.6, the LR statistic, for testing 𝐻 0 : 𝜎 𝜇 2 = 𝜎 𝑣 2 =0, equals , providing highly strong evidence against 𝐻 0 . An LR test for testing 𝐻 0 : 𝜎 𝜇 2 =0, allowing 𝜎 𝑣 2 >0, can be conducted using the output for one-way individual RE and the output for two-way RE. From Table 4.6, the output of two-way RE by Stata command xtmixed with option mle, we obtain Log likelihood = , From Table 4.5, the output of one-way individual RE by Stata commands xtreg and mle, we have Log likelihood = , Therefore, LR=− − = , with df = 1, highly significant against 𝐻 0 . An LR test for testing 𝐻 0 : 𝜎 𝑣 2 =0, allowing 𝜎 𝜇 2 >0, can be conducted using the outputs for one-way time RE and two-way RE.

Wald Tests for Random Effects
Second general test for testing the above hypotheses is the Wald test. It has the following general format, 𝑊= 𝜃 − 𝜃 0 ′ Estimated Var( 𝜃 ) −1 𝜃 − 𝜃 0 , where 𝜃 and 𝜃 0 are the estimated and hypothesized values of 𝜃. under H0, W ~ 𝜒 𝑑𝑓 2 , with df = dimension of the parameter vector 𝜃. See Greene (2012, Sec. 14.6) for details. From Table 4.5, we have 𝜎 𝜇 = with estimated standard error being Therefore, the Wald test for testing 𝐻 0 : 𝜎 𝜇 = 0, given 𝜎 𝑣 =0, has a value W = ( / )2 = 88.65, a super strong evidence against 𝐻 0 . A Wald test for testing 𝐻 0 : 𝜎 𝑣 =0, given 𝜎 𝜇 =0, can be conducted using the outputs for one-way time RE. More tests can be done … .

LM Tests for Random Effects
The third and perhaps the most popular test may be Lagrange multiplier (LM) test or efficient score (or just score) test. It has the following general form: 𝐿𝑀= 𝜕ln𝐿( 𝜃 𝑅 ) 𝜕 𝜃 𝑅 ′ I( 𝜃 𝑅 ) −1 𝜕ln𝐿( 𝜃 𝑅 ) 𝜕 𝜃 𝑅 , where 𝜃 𝑅 is the restricted estimate of the parameter vector 𝜃. under H0, LM ~ 𝜒 𝑑𝑓 2 , with df = dimension of 𝜃. See Greene (2012, Sec. 14.6) for details. Baltagi (2013, Chap. 4) presents several conditional, marginal, and joint LM tests for testing the random effects. There seem no State commands for implementing these tests. However, the LM tests are often very simple, requiring only the OLS residuals, and thus can be easily computed using Stata.

The joint and the two conditional tests for random effects: 𝐿𝑀 𝐽 : joint LM test of 𝐻 0 : 𝜎 𝜇 2 = 𝜎 𝑣 2 =0, 𝐿𝑀 C1 : conditional LM test of 𝐻 0 : 𝜎 𝜇 2 =0, given 𝜎 𝑣 2 =0, 𝐿𝑀 C2 : conditional LM test of 𝐻 0 : 𝜎 𝑣 2 =0, given 𝜎 𝜇 2 =0, can be derived together. They all depend only on the OLS residuals of the regression 𝑦 𝑖𝑡 =𝛼+ 𝑋 𝑖𝑡 ′ 𝛽+ 𝑢 𝑖𝑡 , and take the simple forms: 𝐿𝑀 𝐽 = 𝑁𝑇 2(𝑇−1) 1− 𝑢 ′ ( 𝐼 𝑁 ⊗ 𝐽 𝑇 ) 𝑢 𝑢 ′ 𝑢 𝑁𝑇 2(𝑇−1) 1− 𝑢 ′ ( 𝐽 𝑁 ⊗ 𝐼 𝑇 ) 𝑢 𝑢 ′ 𝑢 2 ≡ 𝐿𝑀 C ≡ 𝐿𝑀 C2 where 𝑢 is the vector of OLS residuals. Their asymptotic null distributions are 𝜒 2 2 , 𝜒 1 2 , and 𝜒 1 2 , respectively. For details on these tests, see Baltagi (2013, Chap. 4). Calculations of these test statistics can be done using Stata, based on the saved values of 𝑢 . There are alternative LM-type of tests; see , e.g., Baltagi (2013, Chap. 4).

Marginal LM test of 𝐻 0 : 𝜎 𝜇 2 =0, allowing 𝜎 𝑣 2 >0 has the form: 𝐿𝑀 M1 = 𝑇 𝜎 𝜎 𝑣 𝑇−1 [ 𝜎 𝑣 4 + (𝑁−1) 𝜎 2 4 ] 𝜎 𝑄 1 −1 + 𝑁−1 𝜎 𝑣 2 𝑄 2 −1 , where 𝑄 1 = 1 𝜎 𝑢 ′ ( 𝐽 𝑁 ⊗ 𝐽 𝑇 ) 𝑢 , 𝑄 2 = 1 (𝑁−1) 𝜎 𝑣 2 𝑢 ′ ( 𝐸 𝑁 ⊗ 𝐽 𝑇 ) 𝑢 , 𝑢 is residuals from GLS on 𝑦 𝑖𝑡 =𝛼+ 𝑋 𝑖𝑡 ′ 𝛽+ 𝜆 𝑡 + 𝑢 𝑖𝑡 ， 𝜎 2 2 = 1 𝑇 𝑢 ′ ( 𝐽 𝑁 ⊗ 𝐼 𝑇 ) 𝑢 , 𝜎 𝑣 2 = 1 𝑇(𝑁−1) 𝑢 ′ 𝐸 𝑁 ⊗ 𝐼 𝑇 𝑢 , and under 𝐻 0 , 𝐿𝑀 M1 ~ 𝜒 1 2 . Marginal LM test of 𝐻 0 : 𝜎 𝑣 2 =0, allowing 𝜎 𝜇 2 >0 has the form: 𝐿𝑀 M2 = 𝑁 𝜎 𝜎 𝑣 𝑁−1 [ 𝜎 𝑣 4 +(𝑇−1) 𝜎 1 4 ] 𝜎 𝑅 1 −1 + 𝑁−1 𝜎 𝑣 2 𝑅 2 −1 , where 𝑅 1 = 1 𝜎 𝑢 ′ ( 𝐽 𝑁 ⊗ 𝐽 𝑇 ) 𝑢 , 𝑅 2 = 1 (𝑇−1) 𝜎 𝑣 2 𝑢 ′ ( 𝐽 𝑁 ⊗ 𝐸 𝑇 ) 𝑢 , 𝑢 is residuals from GLS on 𝑦 𝑖𝑡 =𝛼+ 𝑋 𝑖𝑡 ′ 𝛽+ 𝜇 𝑖 + 𝑢 𝑖𝑡 , 𝜎 1 2 = 1 𝑁 𝑢 ′ ( 𝐼 𝑇 ⊗ 𝐽 𝑇 ) 𝑢 , 𝜎 𝑣 2 = 1 𝑇(𝑁−1) 𝑢 ′ 𝐼 𝑁 ⊗ 𝐸 𝑇 𝑢 , and under 𝐻 0 , 𝐿𝑀 M2 ~ 𝜒 1 2 .

4.4. Hausman Test for RE vs FE
An extensive discussion on fixed effects (FE) vs random effects (RE) was given in Sec In fact, the essential distinction in microeconometric analysis is between FE and RE models. If the effects are fixed, then the pooled OLS and RE estimators are inconsistent, and the within (or FE) estimator needs to be used; The within estimator is otherwise less desirable, because using only within variation leads to less-efficient estimation and inability to estimate effects of time-invariant regressors. Applied researchers therefore face the problem of choosing an appropriate panel data model. Hausman (1978) developed tests helping researchers to choose between an FE model and an RE Model, referred to as Hausman test in the econometrics literature.

Hausman Test for RE vs FE
Consider the panel data model with one-way individual effects: 𝑦 𝑖𝑡 =𝛼+ 𝑋 𝑖𝑡 ′ 𝛽+ 𝑢 𝑖𝑡 , 𝑢 𝑖𝑡 = 𝜇 𝑖 + 𝑣 𝑖𝑡 , i = 1, , N and t = 1, , T. Let 𝛽 RE be the random effect (GLS) estimator, and 𝛽 FE be the fixed effect (within) estimator of 𝛽. If 𝐻 0 :𝐸 𝑢 𝑖𝑡 𝑋 𝑖𝑡 =0, then both 𝛽 RE and 𝛽 FE are unbiased and consistent for 𝛽, but 𝛽 RE is more efficient than 𝛽 FE . If, however, 𝐸 𝑢 𝑖𝑡 𝑋 𝑖𝑡 ≠0 due to 𝐸 𝜇 𝑖 𝑋 𝑖𝑡 ≠0, then only 𝛽 FE is unbiased and consistent for 𝛽 as FE estimation wipes out 𝜇 𝑖 . Hausman (1978) suggests to compare 𝛽 RE and 𝛽 FE by using 𝑞 = 𝛽 RE − 𝛽 FE . Hausman showed that Var( 𝑞 )=Var( 𝛽 FE )−Var( 𝛽 RE ).

Consider the panel data model with one-way individual effects: 𝑦 𝑖𝑡 =𝛼+ 𝑋 𝑖𝑡 ′ 𝛽+ 𝑢 𝑖𝑡 , 𝑢 𝑖𝑡 = 𝜇 𝑖 + 𝑣 𝑖𝑡 , i = 1, , N and t = 1, , T. Let 𝛽 RE be the random effect (GLS) estimator, and 𝛽 FE be the fixed effect (within) estimator of 𝛽. If 𝐻 0 :𝐸 𝑢 𝑖𝑡 𝑋 𝑖𝑡 =0, then both 𝛽 RE and 𝛽 FE are unbiased and consistent for 𝛽, but 𝛽 RE is more efficient than 𝛽 FE . If, however, 𝐸 𝑢 𝑖𝑡 𝑋 𝑖𝑡 ≠0 due to 𝐸 𝜇 𝑖 𝑋 𝑖𝑡 ≠0, then only 𝛽 FE is unbiased and consistent for 𝛽 as FE estimation wipes out 𝜇 𝑖 . Hausman (1978) suggests to compare 𝛽 RE and 𝛽 FE by using 𝑞 1 = 𝛽 RE − 𝛽 FE . Hausman showed: Var 𝑞 1 = Var( 𝛽 FE )−Var 𝛽 RE , in general!

If 𝛽 RE is GLS estimator and 𝛽 FE is within estimator of 𝛽, then, Var 𝑞 1 = 𝜎 𝑣 2 𝑋 ′ 𝑄𝑋 −1 − 𝑋 ′ Ω −1 𝑋 −1 . Hence, the Hausman test statistic is given by 𝑚 1 = 𝑞 1 ′ Var 𝑞 1 −1 𝑞 1 Hausman test still applies if 𝛽 RE and 𝛽 FE are from two-way model with the former being the two-way FE estimator and the latter the two-way RE GLS estimator. The variance of the difference 𝑞 2 = 𝛽 RE − 𝛽 FE , again satisfies Var 𝑞 2 = Var( 𝛽 FE )−Var 𝛽 RE .

Table 4.7. Returns to Schooling Data
Hausman Test -- Stata The Stata command hausman implements the standard form of Hausman test. The sigmamore option specifies that both covariance matrices are based on the (same) estimated disturbance variance from the efficient estimator. Suppose we have already stored the within estimator as FE and the RE estimator as RE. Consider the “Returns to Schooling Data” introduced in Chap. 3. First, issue the following commands to get and store FE and RE estimates Table 4.7. Returns to Schooling Data . quietly xtreg lwage exp expsq wks, fe . estimates store FE . quietly xtreg lwage exp expsq wks, re . estimates store RE

Hausman Test -- Stata Table 4.7. Cont’d
. hausman FE RE, sigmamore ---- Coefficients ---- | (b) (B) (b-B) sqrt(diag(V_b-V_B)) | FE RE Difference S.E. exp | expsq | wks | b = consistent under Ho and Ha; obtained from xtreg B = inconsistent under Ha, efficient under Ho; obtained from xtreg Test: Ho: difference in coefficients not systematic chi2(3) = (b-B)'[(V_b-V_B)^(-1)](b-B) = Prob>chi2 = The overall statistic, chi2(3), has p = This leads to strong rejection of the null hypothesis that RE provides consistent estimates.

Chapter 4: Test Hypotheses with Panel Data

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Chapter 4: Test Hypotheses with Panel Data

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