Formulary Applied Econometrics

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1 Department of Economics Formulary Applied Econometrics c c Seminar of Statistics University of Fribourg

2 Formulary Applied Econometrics 1 Rescaling With y = cy we have: ˆβ = cˆβ With x = Cx we have: ˆβ = C 1 ˆβ Standardized Regression Model with and z y = ˆβ 1 z ˆβ K z K + û z k = (x k x k )/ˆσ k und z y = (y ȳ)/ˆσ y ˆβ k = ˆσ k ˆσ y ˆβk ; k = 1,..., K Logarithmic Transformation With ln(y) = β 0 + β k ln(x k ) we have: % y/% x k β k With ln(y) = β 0 + β k x k we have: % y/ x k = 100 β k With y = β 0 + β k ln(x k ) (x > 0) we have: y/% x k β k /100 Prediction of ŷ by ln(y)-model: ŷ = ˆαexp( lny) With u N(0, σ 2 ) we have: ˆα = exp(ˆσ 2 /2) Goodness-of-fit Total sum of squares: SST = (y n ȳ) 2

3 Formulary Applied Econometrics 2 Regression sum of squares: Sum of squared errors: We have: Variance decomposition: SSR = (ŷ n ȳ) 2 SSE = (y n ŷ n ) 2 SST = SSR + SSE Coefficient of determination: R 2 := s2 ŷ s 2 y s 2 y = s 2 ŷ + s 2 e = 1 s2 e s 2 y = SSR SST = 1 SSE SST Adjusted R 2 : R 2 := 1 SSE/(N K 1) SST/(N 1) = 1 ( ) N 1 (1 R 2 ) N K 1 Binary Independent Variables Semi-elasticity: % y/ x k = 100 [exp( ˆβ k ) 1] F-Statistic F := (SSE r SSE ur )/q SSE ur /(N K 1) F q,n K 1 with SSE r : sum of squared errors of the restricted model and SSE ur : sum of squared errors of the unrestricted model and q = df r df ur Chow-Statistic F = [SSE P (SSE 1 + SSE 2 )] [N 2(K + 1)] SSE 1 + SSE 2 J with SSE 1 = SSE for group 1 with n 1 observations and SSE 2 = SSE for group 2 with n 2 observations and J: number of restrictions F J,N 2K 2 SSE ur = SSE 1 + SSE 2

4 Formulary Applied Econometrics 3 Binary Dependent Variables implies for the variable y y = β 0 + β 1 x β K x K + u E(y x) = β 0 + β 1 x β K x K Response probability: P (y = 1 x) = β 0 + β 1 x β K x K Linear probability model: P (y = 0 x) = 1 P (y = 1 x) Variance of y: Limited dependent variable model: mit 0 < G(z) < 1. Logit model: Probit model: P (y = 1 x) = β k x k V ar(y x) = p(x)[1 p(x)] P (y = 1 x) = G(β 0 + β 1 x β K x K ) = G(β 0 + xβ) ez G(z) = 1 + e z G(z) = Φ(z) = z φ(v)dv where φ(z) is the probability density function (pdf) of the normal distribution. Log-Likelihood function of observation i for the limited dependent variable model: l i (β) = y i log[g(x i β)] + (1 y i )log[1 G(x i β)] The ML-estimator of β maximizes the following log-likelihood function: Likelihood-ratio test: L(β) = l i (β) i=1 Pseudo R-squared: LR = 2(L ur L r ) χ 2 q pseudo R-squared = 1 L ur /L 0 with the log-likelihood function L 0 for the model with only the constant β 0. Average marginal effect of an exogenous variable x k : [ ] 1 g( N ˆβ 0 + x n ˆβ) ˆβ k i=1

5 Formulary Applied Econometrics 4 Quality of the Prediction Root-mean-square error (RMSE): Thiel s U statistic: RMSE = 1 n (y n i ŷ i ) 2 U = i=1 RMSE 1 n n i=1 ŷ2 i + 1 n n i=1 y2 i where n is the number of observations of the prediction sample. Jarque-Bera Test Test statistics and distributions: H 0 : The sample (residuals) is normally distributed. JB = n 6 ( S 2 + ) (K 3)2 χ with skewness S = µ 3 /σ 3 and Kurtosis K = µ 4 /σ 4. For the normal distribution we have: S = 0 and K = 3). Heteroscedasticity V ar(ũ) = σ σn 2, Heteroscedasticity-robust variances For simple regressions: with OLS residuals û 2 n and For multiple regressions: V ar W ( ˆβ k ) = N (x n x) 2 û 2 n SST 2 x SST x = (x n x) 2 V ar W ( ˆβ N ˆr 2 k ) = nkû 2 n SSEk 2 with ˆr nk : n-th residual from the regression of x k on the other variables.

6 Formulary Applied Econometrics 5 Heteroscedasticity-robust t-values: Tests for heteroscedasticity: Breusch-Pagan test: t = ˆβ k V ar W ( ˆβ k ) u 2 = δ 0 + δ 1 x 1 + δ 2 x δ K x K + v White test: H 0 : δ 1 = δ 2 =... = δ K = 0 u 2 =δ 0 + δ 1 x δ K x K + δ K+1 x δ K+Kx 2 K + δ 2K+1 x 1 x δ 2K+K!/((K 2)!2!) x K 1 x K + v or: H 0 : δ 1 = δ 2 =... = δ 2K+K!/((K 2)!2!) x K 1 = 0 u 2 =α 0 + α 1 ŷ + α 2 ŷ 2 + v F-Statistic: F = H 0 : α 1 = α 2 = 0 R 2 û 2 /L (1 R 2 û 2 )/(N L 1) F L,N L 1 with R 2 û 2 : coefficient of determination of the model with squared residuals û 2 and L regressors of the auxiliary regression. LM-Statistic: with L regressors of the auxiliary regression. LM = N R 2 û 2 χ2 L Estimated weighted LS-estimator under heteroscedasticity: Regression model: y = β 0 x 0 + β 1x β Kx K + u with y = 1/ h, x k = x k / h (k = 0,...K) und u = u/ h (x 0 = (1,..., 1)). ĥ = e ln(û 2 ) with ln(û 2 ) from the regression of ln(û 2 ) on x 1,..., x K.

7 Formulary Applied Econometrics 6 Time Series Stochastic process: Statistical model: ỹ t mit t = 1,..., y t = β 0 + β 1 z 1t + β 2 z 2t β K z Kt + u t (t = 1,..., T) Finite distributed lag model of order q (FDL(q)): y t = α 0 + δ 0 z t + δ 1 z t δ q z t q + u t Impact multiplier: IP = δ 0 Long-run multiplier: LRP = Moving-average process of order p (MA(p)): q δ i i=0 x t = e t + α 1 e t 1 + α 2 e t α q e t q, t = 1, 2,..., Autoregressive process of order q (AR(q)): y t = ρ 1 y t 1 + ρ 2 y t ρ q y t q + e t, t = 1, 2,... Autocorrelation Tests for autocorrelation: Null hypothesis: For the AR(1)-model serial correlation: H 0 : ρ = 0 u t = ρu t 1 + e t, t = 2,..., T Durbin-Watson statistic: DW = Tt=2 (û t û t 1 ) 2 Tt=2 û 2 t Null hypothesis for AR(q) serial correlation: H 0 : ρ 1 =... = ρ q = 0 Estimated weighted OLS-estimators under serial correlation: Regression model: ỹ t = β 0 x t0 + β 1 x t β K x tk + error t with ỹ t = y t ρy t 1, x ti = x tk ρx t 1,k (t = 2,..., T; k = 0,..., K) (Cochrane-Orcutt procedure) and ỹ 1 = (1 ρ 2 ) 1/2 y 1, x 1i = (1 ρ 2 ) 1/2 x 1i (Prais-Winston procedure). ρ stems from the regression of the AR(1) process: u t = ρu t 1 + e t, t = 2,..., T.