Econometrics of Panel Data
|
|
- Virgil Blankenship
- 5 years ago
- Views:
Transcription
1 Econometrics of Panel Data Jakub Mućk Meeting # 4 Jakub Mućk Econometrics of Panel Data Meeting # 4 1 / 30
2 Outline 1 Two-way Error Component Model Fixed effects model Random effects model 2 Non-spherical variance-covariance matrix of the error term Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 2 / 30
3 Two-way Error Component Model Two-way Error Component Model y it = α + X itβ + u it i {1,..., N}, t {1,..., T }, (1) the composite error component: u i,t = µ i + λ t + ε i,t, (2) where: µi the unobservable individual-specific effect; λt the unobservable time-specific effect; ui,t the remainder disturbance. Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 3 / 30
4 Fixed effects model Two-way fixed effects model: where y it = α i + λ t + β 1 x 1it β k x kit + u it, (3) αi - the individual-specific intercept; λi - the period-specific intercept; uit N ( 0, σu) 2. As in the case of the one-way fixed effect model, independent variables x 1,..., x k cannot be time invariant (for a given unit). The estimation: 1 the least squares dummy variable estimator (LSDV); 2 the within estimator; 3 combination of the above methods. Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 4 / 30
5 The LSDV estimator In the analogous fashion to the one-way fixed effect model, we define the following dummy variables: for the j-th unit: { 1 i = j D jit = 0 otherwise, (4) for the τ-th period: B τit = { 1 t = τ 0 otherwise. (5) Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 5 / 30
6 The LSDV estimator In the analogous fashion to the one-way fixed effect model, we define the following dummy variables: for the j-th unit: { 1 i = j D jit = 0 otherwise, (4) for the τ-th period: B τit = { 1 t = τ 0 otherwise. (5) The LSDV estimator can be obtained by estimating the following model: N N y it = α j D jit + λ τ B τit + β 1 x 1it β k x kit + u i,t. (6) j=1 τ=1 where u i,t N ( 0, σu) 2. The parameters in equation (7) can be estimated with the OLS. Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 5 / 30
7 The LSDV estimator: test for poolability The LSDV estimator: N N y it = α j D jit + λ τ B τit + β 1 x 1it β k x kit + u i,t, (7) j=1 where u i,t N ( 0, σ 2 u). τ=1 We can test whether the individual-specific and period-specific effects are significantly different: H 0 : α 1 =... = α k λ 1 =... = λ T = 0. (8) To verify the null described by (8) we compare the sum of squared errors from the pooled model with (with restrictions, SSE R ) with the sum of squared errors from the LSDV model (without restrictions, SSE R ). The test statistics F: F = (SSE R SSE U )/(N T 1) SSE U /(NT K T ) if null is emph then F F (N T 1,NT K T ). (9) Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 6 / 30
8 The LSDV estimator: test for poolability The LSDV estimator: N N y it = α j D jit + λ τ B τit + β 1 x 1it β k x kit + u i,t, (7) j=1 where u i,t N ( 0, σ 2 u). τ=1 We can test whether the individual-specific and period-specific effects are significantly different: H 0 : α 1 =... = α k λ 1 =... = λ T = 0. (8) To verify the null described by (8) we compare the sum of squared errors from the pooled model with (with restrictions, SSE R ) with the sum of squared errors from the LSDV model (without restrictions, SSE R ). The test statistics F: F = (SSE R SSE U )/(N T 1) SSE U /(NT K T ) if null is emph then F F (N T 1,NT K T ). One might test only individual or period effects: (9) H 0 : α 1 =... = α k, (10) H 0 : λ 1 =... = λ k. (11) The construction of test for poolability is the same but we use different degrees of freedom. Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 6 / 30
9 The within estimator Consider the following transformations: averaging over time (for each unit i): ȳi, averaging over unit (for each period t): ȳt, averaging over time and unit: ȳ, for the dependent variable y it : ȳ i = 1 T T y it, t ȳ t = 1 N N i y it, ȳ = 1 NT T t N y it. i Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 7 / 30
10 The within estimator Consider the following transformations: averaging over time (for each unit i): ȳi, averaging over unit (for each period t): ȳt, averaging over time and unit: ȳ, for the dependent variable y it : ȳ i = 1 T T y it, t ȳ t = 1 N N i y it, ȳ = 1 NT T t N y it. i Given the above transformations we can eliminate both the individual- and period specific effects: (y it ȳ i ȳ t + ȳ it ) = β (x it x i x t + x it ) + (u it ū i ū t + ū it ). (12) Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 7 / 30
11 The within estimator The within estimator: where ỹ it = β 1 x 1it β k x kit + ũ it (13) ỹ it = y it ȳ i ȳ t + ȳ it x 1it = x 1it x 1i x 1t + x 1it... x kit = x kit x ki x kt + x kit ũ it = u it ū i ū t + ū it. The parameters β 1,..., β k can be estimated by the OLS. The independent variables, i.e., x 1it,..., x kit, cannot be time (unit) invariant. Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 8 / 30
12 Random effects model Two-way random effects model: y it = α + β 1 x 1it β k x kit + u it, (14) where the error component is as follows: where u it = µ i + λ t + ε it, (15) µi is the individual-specific error component and µ i N ( 0, σ 2 µ) ; λt is the period-specific error component and λ t N ( 0, σ 2 λ) ; εit is the idiosyncratic error component and ε it N ( 0, σ 2 ε). Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 9 / 30
13 Random effects model Two-way random effects model: y it = α + β 1 x 1it β k x kit + u it, (14) where the error component is as follows: where u it = µ i + λ t + ε it, (15) µi is the individual-specific error component and µ i N ( 0, σµ) 2 ; λt is the period-specific error component and λ t N ( 0, σλ) 2 ; εit is the idiosyncratic error component and ε it N ( 0, σε) 2. The independent variables can be time invariant. Individual-specific and period-specific effects are independent: E (µ i, µ j ) = 0 if i j, E (λ s, λ t ) = 0 if s t E (µ i, λ t ) = 0 Estimation method: GLS (generalized least squares). Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 9 / 30
14 The two-way RE model - the variance-covariance matrix of error term I It is assumed that the error term in the two-way RE model can be described as follow: u it = µ i + λ t + ε it, (16) where µ i N ( 0, σ 2 µ), λt N ( 0, σ 2 λ) and εit N ( 0, σ 2 ε). The variance-covariance matrix of the error term is not diagonal!. The diagonal elements of the variance covariance matrix of the error term: E ( ) u 2 it = E (µ i + λ t + ε it) 2 = E ( ) µ 2 i + E ( ) λ 2 t + E ( ) ε 2 it + 2cov (µ i, λ t) + 2cov (µ i, ε it) + 2cov (λ t, ε it) }{{}}{{}}{{}}{{}}{{}}{{} =0 =0 =0 =σ 2 µ =σ 2 λ = σ 2 µ + σ 2 λ + σ 2 ε. =σ 2 ε Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 10 / 30
15 The two-way RE model - the variance-covariance matrix of error term II For a given unit, non-diagonal elements of the variance covariance matrix of the error term (t s): cov (u it, u is ) = E [(µ ( i + λ t + ε it ) (µ i + λ s + ε is )], ) = E µ 2 i + E (λ tλ s) + E (ε }{{}}{{} it ε is ) + COV }{{}}{{} 1 =0 =0 =0 = σ 2 µ, =σ 2 µ where COV 1 = cov(λ t, µ i )+cov(λ s, µ i )+cov(λ t, ε it )+cov(λ s, ε it )+2cov(µ i, ε it ). Similarly, for a a given period, non-diagonal elements of the variance covariance matrix of the error term (i j): cov (u it, u jt ) = E [(µ i + λ t + ( ε it (µ j + λ t + ε jt )], ) = E (µ i µ j ) + E λ }{{} 2 t + E (ε it ε jt ) + COV }{{}}{{}}{{} 2 =0 =0 =0 = σ 2 λ. where COV 2 = cov(µ i, λ t )+cov(µ i, λ t )+cov(µ i, ε it )+cov(µ j, ε it )+2cov(λ t, ε it ). =σ 2 λ Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 11 / 30
16 The two-way RE model - the variance-covariance matrix of error term III Finally, the variance-covariance matrix of the error term can be described: σµ 2 + σλ 2 + σ2 ε if i = j, t = s, σµ E (u it, u js ) = 2 if i = j, t s, σλ 2 (17) if i j, t = s, 0 if i j, t s. Unlike the one-way RE model, the variance-covariance matrix E (u it, u js ) is not block-diagonal because there is correlation between units. This correlation is caused by the the period-specific effects and equals σ 2 λ /(σ2 µ + σ 2 λ + σ2 ε). Akin to the one-way RE model there is equicorrelation which is implied by the unit-specific effects and equals σ 2 µ/(σ 2 µ + σ 2 λ + σ2 ε). Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 12 / 30
17 The GLS estimation of the two-way RE model: example Consider the following transformation: zit = z it θ 1 z i θ 2 z t + θ 3 z it, (18) where z it {y it, x 1it,..., x kit, u it } and z i = 1 T T z it, t ȳ t = 1 N N i z it, z = 1 NT T t N z it, i and. θ 1 = 1 θ 2 = 1 θ 3 = θ 1 + θ 2 + σ ε T σ 2 µ + σ 2 ε σ ε Nσ 2 λ + σ 2 ε σ ε T σ 2 µ + Nσ 2 λ + σ2 ε Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 13 / 30 1.
18 The GLS estimation of the two-way RE model: Swamy and Aurora (1972) Swamy and Aurora (1972) propose running three least squares regressions to estimate σ 2 ε, σ 2 µ and σ 2 λ which allow us to calculate θ 1, θ 2 and θ 3. 1 The within regression allows to estimate the variance of the idiosyncratic component, i.e., ˆσ 2 ε. 2 The between (individuals) regression allows to estimate the σ µ which is the estimated variance of the error term from the following regression: ȳ i ȳ = β 1 ( x 1i x 1) β k ( x ki x k ) + ū i ū, and ˆσ I 2 is the estimated variance of the error term from the above regression. Then, the estimated variance of the individual specific-error term can be described as follow ˆσ µ 2 = ˆσ2 I ˆσ ε 2. T 3 The between (periods) regression allows to estimate the σ λ which is the estimated variance of the error term from the following regression: ȳ t ȳ = β 1 ( x 1t x 1) β k ( x kt x k ) + ū t ū, and ˆσ T 2 is the estimated variance of the error term from the above regression. Then, the estimated variance of the individual period-error term can be described as follow ˆσ λ 2 = ˆσ2 T ˆσ ε 2 N. Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 14 / 30
19 The GLS estimation of the two-way RE model: general remarks Alternative estimators of σε, 2 σµ 2 and σλ 2 are proposed by Wallace and Hussain (1969), Amemiya (1971), Fuller and Battese (1974) and Nerlove (1971). In general, estimates of σ 2 ε, σ 2 µ and σ 2 λ allow to calculate θ 1, θ 2 and θ 3 and estimate model for transformed variables: y it = β 1 x 1it β k x kit + u it (19) where z it = z it θ 1 z i θ 2 z t + θ 3 z it and z it {y it, x 1it,..., x kit, u 1it }. Computational warning: sometimes estimates of σµ 2 or σλ 2 could be negative. This is due to the fact that we use two-step strategy rather than jointly estimation. Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 15 / 30
20 Two-way Error Component Model y it = α + β 1 x 1it + β k x kit + µ i + λ t + ε it (20) Random ( effects ) Individual µ i N ( 0, σµ 2 ) time λ t N 0, σλ 2 effects drawn from the random sample = we can estimate the parameters of distribution, i.e., σµ 2 and Fixed effects µ i = individual intercepts α i λ t α i and λ t are assumed to be constant over time σλ 2 Assumptions: (i) E (µ i ε it ) = 0 E (λ t ε it ) = 0 (i) E (µ i ε it ) = 0 E (λ t ε it ) = 0 (ii) E (µ i x it ) = 0 E (λ t x it ) = 0 individual-specific and periodspecific effects are independent of the explanatory variable x it Estimation GLS OLS (within or LSDV) Efficiency higher lower Additional: impossible to use time invariant regressors (collinearity with α i ) Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 16 / 30
21 Outline 1 Two-way Error Component Model Fixed effects model Random effects model 2 Non-spherical variance-covariance matrix of the error term Jakub Mućk Econometrics of Panel Data Non-spherical variance-covariance Meeting # 4 17 / 30
22 Spherical variance-covariance matrix In the previous meeting we ve assumed that the variance covariance matrix of the error term is spherical: E (uu ) = σ 2 ui (21) or, at least, block diagonal (in the RE model): 1 ρ... ρ E (u i. u i.) = Σ i,i = ( σµ 2 + σε 2 ) ρ 1. ρ ρ ρ... 1, (22) where ρ = σ2 µ σµ 2 + σε 2, where σµ 2 and σε 2 stands for the variance of the individual-specific and idiosyncratic error term. Jakub Mućk Econometrics of Panel Data Non-spherical variance-covariance Meeting # 4 18 / 30
23 Non-spherical variance-covariance matrix More general case: E (uu ) = Σ = Σ 1,1 Σ 1,2... Σ 1,N Σ 2,1 Σ 2,2... Σ 2,N Σ N,1 Σ N,2... Σ N,N, (23) where Σ i,j is the variance-covariance matrix of the error term between i-th and j-th (cross-sectional) unit. Jakub Mućk Econometrics of Panel Data Non-spherical variance-covariance Meeting # 4 19 / 30
24 Non-spherical variance-covariance matrix More general case: E (uu ) = Σ = Σ 1,1 Σ 1,2... Σ 1,N Σ 2,1 Σ 2,2... Σ 2,N Σ N,1 Σ N,2... Σ N,N, (23) where Σ i,j is the variance-covariance matrix of the error term between i-th and j-th (cross-sectional) unit. Implications: the least squares estimator is still consistent (if other assumptions are satisfied) but it is no longer BLUE (best linear unbiased estimator). Jakub Mućk Econometrics of Panel Data Non-spherical variance-covariance Meeting # 4 19 / 30
25 The Generalized Least Squares estimator To overcome the problem of non-spherical variance-covariance matrix of the error term If we know Σ and the other assumptions are satisfied one might apply the GLS (Generalized Least Squares) estimator: ˆβ GLS = and the variance-covariance estimator: V ar ( ˆβGLS ) = ( X ˆΣ 1 X) 1 X ˆΣ 1 y, (24) ( X ˆΣ 1 X) 1. (25) In the presence of the non-spherical disturbances the GLS estimator is BLUE. Key challenge: Σ!. Jakub Mućk Econometrics of Panel Data Non-spherical variance-covariance Meeting # 4 20 / 30
26 Non-spherical variance-covariance matrix The variance-covariance matrix of the error term can be non-spherical due to: 1 Autocorrelation (serial correlation). 2 Heteroskedasticity. 3 Cross-sectional dependence. 4 Combination of the above cases. Jakub Mućk Econometrics of Panel Data Non-spherical variance-covariance Meeting # 4 21 / 30
27 Autocorrelation - AR(1) example Autocorrelation or Serial correlation is the correlation of the error term with its past values. AR(1) example: u it = ρu it 1 + η it (26) where ρ (0, 1) and η it N (0, σ 2 η). Autocorrelation signals that disturbances displays some memory. One possible explanation for autocorrelation is that relevant factors are omitted = omitted variables bias. Response to a unit shock in period t = periods ρ = 0.95, ρ = 0.75, ρ = 0.5. Jakub Mućk Econometrics of Panel Data Non-spherical variance-covariance Meeting # 4 22 / 30
28 Detecting autocorrelation Visual inspection: plotting residuals. Simple regressions. Test proposed by Baltagi and Wu (1999). The null hypothesis is about no autocorrelation: H 0 ρ = 0. (27) The test investigates only the first-order autocorrelation. The test statistics is the Durbin-Watson statistic tailored to the panel data. Jakub Mućk Econometrics of Panel Data Non-spherical variance-covariance Meeting # 4 23 / 30
29 The GLS estimator assuming AR(1) error term General idea: 1 The autocorrelation coefficient is estimated from the OLS (within) residuals. 2 All variables are transformed: (1 ˆρ 2 ) 1 2 z it if t = 1 zit = ( ) 1 ( ) (1 ˆρ 2 ) (z ) 2 it 1 ˆρ z ˆρ 2 2 it 1 1 ˆρ if t > 1 2 (28) Note that the above transformation is quite similar to the Prais-Winters transformation. 3 The first observation of each panel should be removed and then it is possible to apply within (FE) estimator to transformed data. 4 Baltagi and Wu propose the GLS estimator of the RE model with the AR error term. The main idea is quite similar to basic RE model. Jakub Mućk Econometrics of Panel Data Non-spherical variance-covariance Meeting # 4 24 / 30
30 Heteroskedasticity Heteroskedasticity refers to the situation in which the variance of the error term is not constant. Example for panel data: when σ 2 u,1... σ 2 u,n. E (uu ) = Σ = diag ( Iσ 2 u1,..., Iσ 2 u,n ) Iσ 2 u, (29) General intuition: uncertainty associated with the outcome y (captured by the variance of the error term) is not constant for various values of independent variables x. Jakub Mućk Econometrics of Panel Data Non-spherical variance-covariance Meeting # 4 25 / 30
31 The Robust standard errors When the error term is heteroskedastic the robust estimator of the variancecovariance can be used to obtain consistent estimates of the standard errors. White s heteroscedasticity-consistent estimator: V ar( ˆβ) = (X X) 1 ( X ˆΣX ) (X X) 1 (30) where ˆΣ = diag(û 2 1,..., û 2 N ). The clustered robust standard errors: All observations are divided into G groups: V ar( ˆβ) = ( X X ) ( G ) 1 (X x iû iû ix i X ) 1. (31) i=1 Jakub Mućk Econometrics of Panel Data Non-spherical variance-covariance Meeting # 4 26 / 30
32 Cross-sectional dependence Cross-sectional dependence takes place when the error terms between individuals at the same time period t are correlated: E (u it u jt ) 0 if i j. (32) The cross-sectional dependence may arise due to the presence of common or/and unobserved factors that become a part of the error term. For instance: common business cycles fluctuations, spillovers, neighborhood effects, herd behavior, and interdependent preferences. Jakub Mućk Econometrics of Panel Data Non-spherical variance-covariance Meeting # 4 27 / 30
33 Testing for the cross-sectional dependence Consider the static panel data model: y it = α + β 1 x 1it β k x kit + u it. (33) The cross-sectional correlation coefficient: T t=1 ˆρ i,j = ûitû jt ( T ) 1 t=1 ûit 2 ( ûjt). (34) 1 T 2 t=1 Note that ρ i,j = ρ j,i. For panel consisting of N unit we get N(N 1)/2 pair-wise correlation coefficients. The hypothesis of interest: The LM statistic (Breusch and Pagan, 1980): H 0 : ρ i,j = 0 if i j. (35) H 1 : ρ i,j 0 if i j (36) LM = T N 1 N i=1 j=i+1 ˆρ 2 i,j. (37) The above statistic is valid for fixed N as T and is asymptomatically distributed as χ 2 with N(N 1)/2 degrees of freedom. The LM statistic exhibits substantial size distortion when N is relatively large due to fact that it is not correctly centered for fixed T. Jakub Mućk Econometrics of Panel Data Non-spherical variance-covariance Meeting # 4 28 / 30
34 Testing for the cross-sectional dependence Pesaran (2004) proposes the following test statistic: N 1 2T N CD = N (N 1) i=1 j=i+1 ˆρ i,j. (38) Under the null about no cross sectional dependence, the CD statistic is normally distributed, i.e., CD N (0, 1), for N and sufficiently large T. The CD statistic can be used in a wide range of panel-data models, e.g., basic static models, homogeneous/heterogeneous dynamic model, nonstationary model. Unbalanced panels: N 1 2 CD = N (N 1) N i=1 j=i+1 Tij ˆρ i,j. (39) Jakub Mućk Econometrics of Panel Data Non-spherical variance-covariance Meeting # 4 29 / 30
35 Testing for the cross-sectional dependence Friedman s statistics: T 1 F R = (40) (N 1)R ave + 1 where R ave is the average Spearman s cross-sectional correlation. The F R statistic is asymptotically χ 2 distributed with with T 1 degrees of freedom, for fixed T and and sufficiently large N. Jakub Mućk Econometrics of Panel Data Non-spherical variance-covariance Meeting # 4 30 / 30
36 Testing for the cross-sectional dependence Friedman s statistics: T 1 F R = (40) (N 1)R ave + 1 where R ave is the average Spearman s cross-sectional correlation. The F R statistic is asymptotically χ 2 distributed with with T 1 degrees of freedom, for fixed T and and sufficiently large N. Frees statistics bases on the sum of the squared rank correlation coefficients R 2 ave: where F RE = N N 1 Rave 2 2 = N (N 1) ( Rave 2 1 ), (41) T 1 N i=1 j=i+1 Tij ˆr i,j. (42) where r i,j is the Spearman s correlation between residuals for i and j unit. The null is rejected when R 2 ave > 1/(T 1) + Q b /N, where Q b is the b-th quantile of the Q distribution (the Q distribution is the weighted sum ot two χ 2 random variables). Both Friedman s and Frees statistic are designed to static panel data models. Jakub Mućk Econometrics of Panel Data Non-spherical variance-covariance Meeting # 4 30 / 30
Econometrics of Panel Data
Econometrics of Panel Data Jakub Mućk Meeting # 4 Jakub Mućk Econometrics of Panel Data Meeting # 4 1 / 26 Outline 1 Two-way Error Component Model Fixed effects model Random effects model 2 Hausman-Taylor
More informationEconometrics of Panel Data
Econometrics of Panel Data Jakub Mućk Meeting # 2 Jakub Mućk Econometrics of Panel Data Meeting # 2 1 / 26 Outline 1 Fixed effects model The Least Squares Dummy Variable Estimator The Fixed Effect (Within
More informationEconometrics of Panel Data
Econometrics of Panel Data Jakub Mućk Meeting # 3 Jakub Mućk Econometrics of Panel Data Meeting # 3 1 / 21 Outline 1 Fixed or Random Hausman Test 2 Between Estimator 3 Coefficient of determination (R 2
More informationEconometrics of Panel Data
Econometrics of Panel Data Jakub Mućk Meeting # 1 Jakub Mućk Econometrics of Panel Data Meeting # 1 1 / 31 Outline 1 Course outline 2 Panel data Advantages of Panel Data Limitations of Panel Data 3 Pooled
More informationEconometrics of Panel Data
Econometrics of Panel Data Jakub Mućk Meeting # 6 Jakub Mućk Econometrics of Panel Data Meeting # 6 1 / 36 Outline 1 The First-Difference (FD) estimator 2 Dynamic panel data models 3 The Anderson and Hsiao
More informationAdvanced Econometrics
Based on the textbook by Verbeek: A Guide to Modern Econometrics Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna May 16, 2013 Outline Univariate
More informationHeteroskedasticity and Autocorrelation
Lesson 7 Heteroskedasticity and Autocorrelation Pilar González and Susan Orbe Dpt. Applied Economics III (Econometrics and Statistics) Pilar González and Susan Orbe OCW 2014 Lesson 7. Heteroskedasticity
More informationEconometrics. Week 4. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague
Econometrics Week 4 Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Fall 2012 1 / 23 Recommended Reading For the today Serial correlation and heteroskedasticity in
More informationRepeated observations on the same cross-section of individual units. Important advantages relative to pure cross-section data
Panel data Repeated observations on the same cross-section of individual units. Important advantages relative to pure cross-section data - possible to control for some unobserved heterogeneity - possible
More informationPanel Data Model (January 9, 2018)
Ch 11 Panel Data Model (January 9, 2018) 1 Introduction Data sets that combine time series and cross sections are common in econometrics For example, the published statistics of the OECD contain numerous
More informationECON 4551 Econometrics II Memorial University of Newfoundland. Panel Data Models. Adapted from Vera Tabakova s notes
ECON 4551 Econometrics II Memorial University of Newfoundland Panel Data Models Adapted from Vera Tabakova s notes 15.1 Grunfeld s Investment Data 15.2 Sets of Regression Equations 15.3 Seemingly Unrelated
More informationPanel Data Models. Chapter 5. Financial Econometrics. Michael Hauser WS17/18 1 / 63
1 / 63 Panel Data Models Chapter 5 Financial Econometrics Michael Hauser WS17/18 2 / 63 Content Data structures: Times series, cross sectional, panel data, pooled data Static linear panel data models:
More informationDiagnostics of Linear Regression
Diagnostics of Linear Regression Junhui Qian October 7, 14 The Objectives After estimating a model, we should always perform diagnostics on the model. In particular, we should check whether the assumptions
More informationLecture 4: Heteroskedasticity
Lecture 4: Heteroskedasticity Econometric Methods Warsaw School of Economics (4) Heteroskedasticity 1 / 24 Outline 1 What is heteroskedasticity? 2 Testing for heteroskedasticity White Goldfeld-Quandt Breusch-Pagan
More informationF9 F10: Autocorrelation
F9 F10: Autocorrelation Feng Li Department of Statistics, Stockholm University Introduction In the classic regression model we assume cov(u i, u j x i, x k ) = E(u i, u j ) = 0 What if we break the assumption?
More informationIntermediate Econometrics
Intermediate Econometrics Heteroskedasticity Text: Wooldridge, 8 July 17, 2011 Heteroskedasticity Assumption of homoskedasticity, Var(u i x i1,..., x ik ) = E(u 2 i x i1,..., x ik ) = σ 2. That is, the
More informationHeteroskedasticity. Part VII. Heteroskedasticity
Part VII Heteroskedasticity As of Oct 15, 2015 1 Heteroskedasticity Consequences Heteroskedasticity-robust inference Testing for Heteroskedasticity Weighted Least Squares (WLS) Feasible generalized Least
More informationEconometric Methods for Panel Data
Based on the books by Baltagi: Econometric Analysis of Panel Data and by Hsiao: Analysis of Panel Data Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies
More informationChapter 6. Panel Data. Joan Llull. Quantitative Statistical Methods II Barcelona GSE
Chapter 6. Panel Data Joan Llull Quantitative Statistical Methods II Barcelona GSE Introduction Chapter 6. Panel Data 2 Panel data The term panel data refers to data sets with repeated observations over
More informationEconometrics Summary Algebraic and Statistical Preliminaries
Econometrics Summary Algebraic and Statistical Preliminaries Elasticity: The point elasticity of Y with respect to L is given by α = ( Y/ L)/(Y/L). The arc elasticity is given by ( Y/ L)/(Y/L), when L
More informationApplied Econometrics (QEM)
Applied Econometrics (QEM) The Simple Linear Regression Model based on Prinicples of Econometrics Jakub Mućk Department of Quantitative Economics Jakub Mućk Applied Econometrics (QEM) Meeting #2 The Simple
More informationIris Wang.
Chapter 10: Multicollinearity Iris Wang iris.wang@kau.se Econometric problems Multicollinearity What does it mean? A high degree of correlation amongst the explanatory variables What are its consequences?
More informationPanel Data Models. James L. Powell Department of Economics University of California, Berkeley
Panel Data Models James L. Powell Department of Economics University of California, Berkeley Overview Like Zellner s seemingly unrelated regression models, the dependent and explanatory variables for panel
More informationEconometrics Multiple Regression Analysis: Heteroskedasticity
Econometrics Multiple Regression Analysis: João Valle e Azevedo Faculdade de Economia Universidade Nova de Lisboa Spring Semester João Valle e Azevedo (FEUNL) Econometrics Lisbon, April 2011 1 / 19 Properties
More informationSensitivity of GLS estimators in random effects models
of GLS estimators in random effects models Andrey L. Vasnev (University of Sydney) Tokyo, August 4, 2009 1 / 19 Plan Plan Simulation studies and estimators 2 / 19 Simulation studies Plan Simulation studies
More informationEcon 582 Fixed Effects Estimation of Panel Data
Econ 582 Fixed Effects Estimation of Panel Data Eric Zivot May 28, 2012 Panel Data Framework = x 0 β + = 1 (individuals); =1 (time periods) y 1 = X β ( ) ( 1) + ε Main question: Is x uncorrelated with?
More informationEstimation of Time-invariant Effects in Static Panel Data Models
Estimation of Time-invariant Effects in Static Panel Data Models M. Hashem Pesaran University of Southern California, and Trinity College, Cambridge Qiankun Zhou University of Southern California September
More informationApplied Microeconometrics (L5): Panel Data-Basics
Applied Microeconometrics (L5): Panel Data-Basics Nicholas Giannakopoulos University of Patras Department of Economics ngias@upatras.gr November 10, 2015 Nicholas Giannakopoulos (UPatras) MSc Applied Economics
More informationTest of hypotheses with panel data
Stochastic modeling in economics and finance November 4, 2015 Contents 1 Test for poolability of the data 2 Test for individual and time effects 3 Hausman s specification test 4 Case study Contents Test
More informationØkonomisk Kandidateksamen 2004 (I) Econometrics 2. Rettevejledning
Økonomisk Kandidateksamen 2004 (I) Econometrics 2 Rettevejledning This is a closed-book exam (uden hjælpemidler). Answer all questions! The group of questions 1 to 4 have equal weight. Within each group,
More informationChristopher Dougherty London School of Economics and Political Science
Introduction to Econometrics FIFTH EDITION Christopher Dougherty London School of Economics and Political Science OXFORD UNIVERSITY PRESS Contents INTRODU CTION 1 Why study econometrics? 1 Aim of this
More informationEmpirical Economic Research, Part II
Based on the text book by Ramanathan: Introductory Econometrics Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna December 7, 2011 Outline Introduction
More informationMultiple Equation GMM with Common Coefficients: Panel Data
Multiple Equation GMM with Common Coefficients: Panel Data Eric Zivot Winter 2013 Multi-equation GMM with common coefficients Example (panel wage equation) 69 = + 69 + + 69 + 1 80 = + 80 + + 80 + 2 Note:
More informationEconometrics of Panel Data
Econometrics of Panel Data Jakub Mućk Meeting # 9 Jakub Mućk Econometrics of Panel Data Meeting # 9 1 / 22 Outline 1 Time series analysis Stationarity Unit Root Tests for Nonstationarity 2 Panel Unit Root
More informationTopic 10: Panel Data Analysis
Topic 10: Panel Data Analysis Advanced Econometrics (I) Dong Chen School of Economics, Peking University 1 Introduction Panel data combine the features of cross section data time series. Usually a panel
More informationAUTOCORRELATION. Phung Thanh Binh
AUTOCORRELATION Phung Thanh Binh OUTLINE Time series Gauss-Markov conditions The nature of autocorrelation Causes of autocorrelation Consequences of autocorrelation Detecting autocorrelation Remedial measures
More informationNonstationary Panels
Nonstationary Panels Based on chapters 12.4, 12.5, and 12.6 of Baltagi, B. (2005): Econometric Analysis of Panel Data, 3rd edition. Chichester, John Wiley & Sons. June 3, 2009 Agenda 1 Spurious Regressions
More informationSpecification testing in panel data models estimated by fixed effects with instrumental variables
Specification testing in panel data models estimated by fixed effects wh instrumental variables Carrie Falls Department of Economics Michigan State Universy Abstract I show that a handful of the regressions
More informationOutline. Nature of the Problem. Nature of the Problem. Basic Econometrics in Transportation. Autocorrelation
1/30 Outline Basic Econometrics in Transportation Autocorrelation Amir Samimi What is the nature of autocorrelation? What are the theoretical and practical consequences of autocorrelation? Since the assumption
More informationEconometrics. Week 6. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague
Econometrics Week 6 Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Fall 2012 1 / 21 Recommended Reading For the today Advanced Panel Data Methods. Chapter 14 (pp.
More informationEconometrics. 9) Heteroscedasticity and autocorrelation
30C00200 Econometrics 9) Heteroscedasticity and autocorrelation Timo Kuosmanen Professor, Ph.D. http://nomepre.net/index.php/timokuosmanen Today s topics Heteroscedasticity Possible causes Testing for
More informationLinear Regression with Time Series Data
Econometrics 2 Linear Regression with Time Series Data Heino Bohn Nielsen 1of21 Outline (1) The linear regression model, identification and estimation. (2) Assumptions and results: (a) Consistency. (b)
More informationEconometrics I KS. Module 2: Multivariate Linear Regression. Alexander Ahammer. This version: April 16, 2018
Econometrics I KS Module 2: Multivariate Linear Regression Alexander Ahammer Department of Economics Johannes Kepler University of Linz This version: April 16, 2018 Alexander Ahammer (JKU) Module 2: Multivariate
More informationIntroductory Econometrics
Based on the textbook by Wooldridge: : A Modern Approach Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna December 11, 2012 Outline Heteroskedasticity
More informationQuantitative Analysis of Financial Markets. Summary of Part II. Key Concepts & Formulas. Christopher Ting. November 11, 2017
Summary of Part II Key Concepts & Formulas Christopher Ting November 11, 2017 christopherting@smu.edu.sg http://www.mysmu.edu/faculty/christophert/ Christopher Ting 1 of 16 Why Regression Analysis? Understand
More informationThe Two-way Error Component Regression Model
The Two-way Error Component Regression Model Badi H. Baltagi October 26, 2015 Vorisek Jan Highlights Introduction Fixed effects Random effects Maximum likelihood estimation Prediction Example(s) Two-way
More informationPanel Data. March 2, () Applied Economoetrics: Topic 6 March 2, / 43
Panel Data March 2, 212 () Applied Economoetrics: Topic March 2, 212 1 / 43 Overview Many economic applications involve panel data. Panel data has both cross-sectional and time series aspects. Regression
More informationDealing With Endogeneity
Dealing With Endogeneity Junhui Qian December 22, 2014 Outline Introduction Instrumental Variable Instrumental Variable Estimation Two-Stage Least Square Estimation Panel Data Endogeneity in Econometrics
More informationFormulary Applied Econometrics
Department of Economics Formulary Applied Econometrics c c Seminar of Statistics University of Fribourg Formulary Applied Econometrics 1 Rescaling With y = cy we have: ˆβ = cˆβ With x = Cx we have: ˆβ
More informationPlease discuss each of the 3 problems on a separate sheet of paper, not just on a separate page!
Econometrics - Exam May 11, 2011 1 Exam Please discuss each of the 3 problems on a separate sheet of paper, not just on a separate page! Problem 1: (15 points) A researcher has data for the year 2000 from
More informationy it = α i + β 0 ix it + ε it (0.1) The panel data estimators for the linear model are all standard, either the application of OLS or GLS.
0.1. Panel Data. Suppose we have a panel of data for groups (e.g. people, countries or regions) i =1, 2,..., N over time periods t =1, 2,..., T on a dependent variable y it and a kx1 vector of independent
More informationRecent Advances in the Field of Trade Theory and Policy Analysis Using Micro-Level Data
Recent Advances in the Field of Trade Theory and Policy Analysis Using Micro-Level Data July 2012 Bangkok, Thailand Cosimo Beverelli (World Trade Organization) 1 Content a) Classical regression model b)
More informationThe Multiple Regression Model Estimation
Lesson 5 The Multiple Regression Model Estimation Pilar González and Susan Orbe Dpt Applied Econometrics III (Econometrics and Statistics) Pilar González and Susan Orbe OCW 2014 Lesson 5 Regression model:
More information1 Estimation of Persistent Dynamic Panel Data. Motivation
1 Estimation of Persistent Dynamic Panel Data. Motivation Consider the following Dynamic Panel Data (DPD) model y it = y it 1 ρ + x it β + µ i + v it (1.1) with i = {1, 2,..., N} denoting the individual
More informationApplied Econometrics (QEM)
Applied Econometrics (QEM) based on Prinicples of Econometrics Jakub Mućk Department of Quantitative Economics Jakub Mućk Applied Econometrics (QEM) Meeting #3 1 / 42 Outline 1 2 3 t-test P-value Linear
More informationEconometrics I Lecture 3: The Simple Linear Regression Model
Econometrics I Lecture 3: The Simple Linear Regression Model Mohammad Vesal Graduate School of Management and Economics Sharif University of Technology 44716 Fall 1397 1 / 32 Outline Introduction Estimating
More informationMultiple Regression Analysis
Multiple Regression Analysis y = 0 + 1 x 1 + x +... k x k + u 6. Heteroskedasticity What is Heteroskedasticity?! Recall the assumption of homoskedasticity implied that conditional on the explanatory variables,
More informationLinear Panel Data Models
Linear Panel Data Models Michael R. Roberts Department of Finance The Wharton School University of Pennsylvania October 5, 2009 Michael R. Roberts Linear Panel Data Models 1/56 Example First Difference
More informationSpatial Regression. 14. Spatial Panels (2) Luc Anselin. Copyright 2017 by Luc Anselin, All Rights Reserved
Spatial Regression 14. Spatial Panels (2) Luc Anselin http://spatial.uchicago.edu 1 fixed effects models random effects models ML estimation IV/2SLS estimation GM estimation specification tests 2 Fixed
More informationSUR Estimation of Error Components Models With AR(1) Disturbances and Unobserved Endogenous Effects
SUR Estimation of Error Components Models With AR(1) Disturbances and Unobserved Endogenous Effects Peter Egger November 27, 2001 Abstract Thispaperfocussesontheestimationoferrorcomponentsmodels in the
More informationUNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS
UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Exam: ECON3150/ECON4150 Introductory Econometrics Date of exam: Wednesday, May 15, 013 Grades are given: June 6, 013 Time for exam: :30 p.m. 5:30 p.m. The problem
More informationEconometrics. Week 8. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague
Econometrics Week 8 Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Fall 2012 1 / 25 Recommended Reading For the today Instrumental Variables Estimation and Two Stage
More informationLECTURE 11. Introduction to Econometrics. Autocorrelation
LECTURE 11 Introduction to Econometrics Autocorrelation November 29, 2016 1 / 24 ON PREVIOUS LECTURES We discussed the specification of a regression equation Specification consists of choosing: 1. correct
More informationEcon 510 B. Brown Spring 2014 Final Exam Answers
Econ 510 B. Brown Spring 2014 Final Exam Answers Answer five of the following questions. You must answer question 7. The question are weighted equally. You have 2.5 hours. You may use a calculator. Brevity
More informationAdvanced Quantitative Methods: panel data
Advanced Quantitative Methods: Panel data University College Dublin 17 April 2012 1 2 3 4 5 Outline 1 2 3 4 5 Panel data Country Year GDP Population Ireland 1990 80 3.4 Ireland 1991 84 3.5 Ireland 1992
More informationPanel Data: Linear Models
Panel Data: Linear Models Laura Magazzini University of Verona laura.magazzini@univr.it http://dse.univr.it/magazzini Laura Magazzini (@univr.it) Panel Data: Linear Models 1 / 45 Introduction Outline What
More informationDirected Tests of No Cross-Sectional Correlation in Large-N Panel Data Models
Directed ests of o Cross-Sectional Correlation in Large- Panel Data Models Matei Demetrescu Christian Albrechts-University of Kiel Ulrich Homm University of Bonn Revised version: April 9, 205 Abstract
More informationDynamic Panel Data Workshop. Yongcheol Shin, University of York University of Melbourne
Dynamic Panel Data Workshop Yongcheol Shin, University of York University of Melbourne 10-12 June 2014 2 Contents 1 Introduction 11 11 Models For Pooled Time Series 12 111 Classical regression model 13
More information10 Panel Data. Andrius Buteikis,
10 Panel Data Andrius Buteikis, andrius.buteikis@mif.vu.lt http://web.vu.lt/mif/a.buteikis/ Introduction Panel data combines cross-sectional and time series data: the same individuals (persons, firms,
More informationCh.10 Autocorrelated Disturbances (June 15, 2016)
Ch10 Autocorrelated Disturbances (June 15, 2016) In a time-series linear regression model setting, Y t = x tβ + u t, t = 1, 2,, T, (10-1) a common problem is autocorrelation, or serial correlation of the
More informationHeteroskedasticity. We now consider the implications of relaxing the assumption that the conditional
Heteroskedasticity We now consider the implications of relaxing the assumption that the conditional variance V (u i x i ) = σ 2 is common to all observations i = 1,..., In many applications, we may suspect
More informationLecture 4: Linear panel models
Lecture 4: Linear panel models Luc Behaghel PSE February 2009 Luc Behaghel (PSE) Lecture 4 February 2009 1 / 47 Introduction Panel = repeated observations of the same individuals (e.g., rms, workers, countries)
More informationLecture 5: Omitted Variables, Dummy Variables and Multicollinearity
Lecture 5: Omitted Variables, Dummy Variables and Multicollinearity R.G. Pierse 1 Omitted Variables Suppose that the true model is Y i β 1 + β X i + β 3 X 3i + u i, i 1,, n (1.1) where β 3 0 but that the
More informationGraduate Econometrics Lecture 4: Heteroskedasticity
Graduate Econometrics Lecture 4: Heteroskedasticity Department of Economics University of Gothenburg November 30, 2014 1/43 and Autocorrelation Consequences for OLS Estimator Begin from the linear model
More informationEconomics 582 Random Effects Estimation
Economics 582 Random Effects Estimation Eric Zivot May 29, 2013 Random Effects Model Hence, the model can be re-written as = x 0 β + + [x ] = 0 (no endogeneity) [ x ] = = + x 0 β + + [x ] = 0 [ x ] = 0
More informationPANEL DATA RANDOM AND FIXED EFFECTS MODEL. Professor Menelaos Karanasos. December Panel Data (Institute) PANEL DATA December / 1
PANEL DATA RANDOM AND FIXED EFFECTS MODEL Professor Menelaos Karanasos December 2011 PANEL DATA Notation y it is the value of the dependent variable for cross-section unit i at time t where i = 1,...,
More informationWeek 11 Heteroskedasticity and Autocorrelation
Week 11 Heteroskedasticity and Autocorrelation İnsan TUNALI Econ 511 Econometrics I Koç University 27 November 2018 Lecture outline 1. OLS and assumptions on V(ε) 2. Violations of V(ε) σ 2 I: 1. Heteroskedasticity
More informationCh 2: Simple Linear Regression
Ch 2: Simple Linear Regression 1. Simple Linear Regression Model A simple regression model with a single regressor x is y = β 0 + β 1 x + ɛ, where we assume that the error ɛ is independent random component
More informationMultiple Regression Analysis: Heteroskedasticity
Multiple Regression Analysis: Heteroskedasticity y = β 0 + β 1 x 1 + β x +... β k x k + u Read chapter 8. EE45 -Chaiyuth Punyasavatsut 1 topics 8.1 Heteroskedasticity and OLS 8. Robust estimation 8.3 Testing
More informationFinQuiz Notes
Reading 10 Multiple Regression and Issues in Regression Analysis 2. MULTIPLE LINEAR REGRESSION Multiple linear regression is a method used to model the linear relationship between a dependent variable
More informationLeast Squares Estimation-Finite-Sample Properties
Least Squares Estimation-Finite-Sample Properties Ping Yu School of Economics and Finance The University of Hong Kong Ping Yu (HKU) Finite-Sample 1 / 29 Terminology and Assumptions 1 Terminology and Assumptions
More informationLinear dynamic panel data models
Linear dynamic panel data models Laura Magazzini University of Verona L. Magazzini (UniVR) Dynamic PD 1 / 67 Linear dynamic panel data models Dynamic panel data models Notation & Assumptions One of the
More informationG. S. Maddala Kajal Lahiri. WILEY A John Wiley and Sons, Ltd., Publication
G. S. Maddala Kajal Lahiri WILEY A John Wiley and Sons, Ltd., Publication TEMT Foreword Preface to the Fourth Edition xvii xix Part I Introduction and the Linear Regression Model 1 CHAPTER 1 What is Econometrics?
More informationApplied Statistics and Econometrics
Applied Statistics and Econometrics Lecture 6 Saul Lach September 2017 Saul Lach () Applied Statistics and Econometrics September 2017 1 / 53 Outline of Lecture 6 1 Omitted variable bias (SW 6.1) 2 Multiple
More informationChapter 15 Panel Data Models. Pooling Time-Series and Cross-Section Data
Chapter 5 Panel Data Models Pooling Time-Series and Cross-Section Data Sets of Regression Equations The topic can be introduced wh an example. A data set has 0 years of time series data (from 935 to 954)
More informationWeighted Least Squares
Weighted Least Squares The standard linear model assumes that Var(ε i ) = σ 2 for i = 1,..., n. As we have seen, however, there are instances where Var(Y X = x i ) = Var(ε i ) = σ2 w i. Here w 1,..., w
More informationthe error term could vary over the observations, in ways that are related
Heteroskedasticity We now consider the implications of relaxing the assumption that the conditional variance Var(u i x i ) = σ 2 is common to all observations i = 1,..., n In many applications, we may
More informationHeteroskedasticity. in the Error Component Model
Heteroskedasticity in the Error Component Model Baltagi Textbook Chapter 5 Mozhgan Raeisian Parvari (0.06.010) Content Introduction Cases of Heteroskedasticity Adaptive heteroskedastic estimators (EGLS,
More informationeconstor Make Your Publications Visible.
econstor Make Your Publications Visible. A Service of Wirtschaft Centre zbwleibniz-informationszentrum Economics Born, Benjamin; Breitung, Jörg Conference Paper Testing for Serial Correlation in Fixed-Effects
More informationReview of Econometrics
Review of Econometrics Zheng Tian June 5th, 2017 1 The Essence of the OLS Estimation Multiple regression model involves the models as follows Y i = β 0 + β 1 X 1i + β 2 X 2i + + β k X ki + u i, i = 1,...,
More informationEC327: Advanced Econometrics, Spring 2007
EC327: Advanced Econometrics, Spring 2007 Wooldridge, Introductory Econometrics (3rd ed, 2006) Chapter 14: Advanced panel data methods Fixed effects estimators We discussed the first difference (FD) model
More informationInternal vs. external validity. External validity. This section is based on Stock and Watson s Chapter 9.
Section 7 Model Assessment This section is based on Stock and Watson s Chapter 9. Internal vs. external validity Internal validity refers to whether the analysis is valid for the population and sample
More informationECON 4230 Intermediate Econometric Theory Exam
ECON 4230 Intermediate Econometric Theory Exam Multiple Choice (20 pts). Circle the best answer. 1. The Classical assumption of mean zero errors is satisfied if the regression model a) is linear in the
More informationHeteroscedasticity. Jamie Monogan. Intermediate Political Methodology. University of Georgia. Jamie Monogan (UGA) Heteroscedasticity POLS / 11
Heteroscedasticity Jamie Monogan University of Georgia Intermediate Political Methodology Jamie Monogan (UGA) Heteroscedasticity POLS 7014 1 / 11 Objectives By the end of this meeting, participants should
More informationTopic 7: Heteroskedasticity
Topic 7: Heteroskedasticity Advanced Econometrics (I Dong Chen School of Economics, Peking University Introduction If the disturbance variance is not constant across observations, the regression is heteroskedastic
More informationSection 2 NABE ASTEF 65
Section 2 NABE ASTEF 65 Econometric (Structural) Models 66 67 The Multiple Regression Model 68 69 Assumptions 70 Components of Model Endogenous variables -- Dependent variables, values of which are determined
More informationAuto correlation 2. Note: In general we can have AR(p) errors which implies p lagged terms in the error structure, i.e.,
1 Motivation Auto correlation 2 Autocorrelation occurs when what happens today has an impact on what happens tomorrow, and perhaps further into the future This is a phenomena mainly found in time-series
More informationLecture 3: Multiple Regression
Lecture 3: Multiple Regression R.G. Pierse 1 The General Linear Model Suppose that we have k explanatory variables Y i = β 1 + β X i + β 3 X 3i + + β k X ki + u i, i = 1,, n (1.1) or Y i = β j X ji + u
More informationHOW IS GENERALIZED LEAST SQUARES RELATED TO WITHIN AND BETWEEN ESTIMATORS IN UNBALANCED PANEL DATA?
HOW IS GENERALIZED LEAST SQUARES RELATED TO WITHIN AND BETWEEN ESTIMATORS IN UNBALANCED PANEL DATA? ERIK BIØRN Department of Economics University of Oslo P.O. Box 1095 Blindern 0317 Oslo Norway E-mail:
More informationThree Essays on Testing for Cross-Sectional Dependence and Specification in Large Panel Data Models
Syracuse University SURFACE Dissertations - ALL SURFACE July 06 Three Essays on Testing for Cross-Sectional Dependence and Specification in Large Panel Data Models Bin Peng Syracuse University Follow this
More information