Econometrics-I-15

# Godfreys lm test regression of et on et 1 and xt uses

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Godfrey’s LM test. Regression of et on et-1 and xt . Uses a “partial correlation.” ™  20/45

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Part 15: Generalized Regression Applications Consumption “Function” Log real consumption vs. Log real disposable income ( Aggregate U.S. Data, 1950I – 2000IV. Table F5.2 from text) ---------------------------------------------------------------------- Ordinary least squares regression ............ LHS=LOGC Mean = 7.88005 Standard deviation = .51572 Number of observs. = 204 Model size Parameters = 2 Degrees of freedom = 202 Residuals Sum of squares = .09521 Standard error of e = .02171 Fit R-squared = .99824 <<<*** Adjusted R-squared = .99823 Model test F[ 1, 202] (prob) =114351.2(.0000) --------+------------------------------------------------------------- Variable| Coefficient Standard Error t-ratio P[|T|>t] Mean of X --------+------------------------------------------------------------- Constant| -.13526*** .02375 -5.695 .0000 LOGY| 1.00306*** .00297 338.159 .0000 7.99083 --------+------------------------------------------------------------- ™  21/45
Part 15: Generalized Regression Applications Least Squares Residuals: r = .91 ™  22/45

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Part 15: Generalized Regression Applications Conventional vs. Newey-West +---------+--------------+----------------+--------+---------+----------+ |Variable | Coefficient | Standard Error |t-ratio |P[|T|>t] | Mean of X| +---------+--------------+----------------+--------+---------+----------+ Constant -.13525584 .02375149 -5.695 .0000 LOGY 1.00306313 .00296625 338.159 .0000 7.99083133 +---------+--------------+----------------+--------+---------+----------+ |Newey-West Robust Covariance Matrix |Variable | Coefficient | Standard Error |t-ratio |P[|T|>t] | Mean of X| +---------+--------------+----------------+--------+---------+----------+ Constant -.13525584 .07257279 -1.864 .0638 LOGY 1.00306313 .00938791 106.846 .0000 7.99083133 ™  23/45
Part 15: Generalized Regression Applications FGLS +---------------------------------------------+ | AR(1) Model: e(t) = rho * e(t-1) + u(t) | | Initial value of rho = .90693 | <<<*** | Maximum iterations = 100 | | Method = Prais - Winsten | | Iter= 1, SS= .017, Log-L= 666.519353 | | Iter= 2, SS= .017, Log-L= 666.573544 | | Final value of Rho = .910496 | <<<*** | Iter= 2, SS= .017, Log-L= 666.573544 | | Durbin-Watson: e(t) = .179008 | | Std. Deviation: e(t) = .022308 | | Std. Deviation: u(t) = .009225 | | Durbin-Watson: u(t) = 2.512611 | | Autocorrelation: u(t) = -.256306 | | N[0,1] used for significance levels | +---------------------------------------------+ +---------+--------------+----------------+--------+---------+----------+ |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X| +---------+--------------+----------------+--------+---------+----------+ Constant -.08791441 .09678008 -.908 .3637 LOGY .99749200 .01208806 82.519 .0000 7.99083133 RHO .91049600 .02902326 31.371 .0000 ™  24/45

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Part 15: Generalized Regression Applications Seemingly Unrelated Regressions The classical regression model, y i = Xii + i. Applies to each of M equations and T observations. Familiar example: The capital asset pricing model: ( r m - r f) = m i + m( r market – r f ) + m Not quite the same as a panel data model. M is usually small - say 3 or 4. (The CAPM might have M in the thousands, but it is a special case for other reasons.) ™  25/45
Part 15: Generalized Regression Applications Formulation Consider an extension of the groupwise heteroscedastic model: We had yi = Xi + i with E[ i|X ] = 0, Var[ i|X ] = i2 I . Now, allow two extensions: Different coefficient vectors for each group, Correlation across the observations at each specific point in time (think about the CAPM above. Variation in excess returns is affected both by firm specific factors and by the economy as a whole). Stack the equations to obtain a GR model. ™  26/45

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Part 15: Generalized Regression Applications SUR Model ™  27/45 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 2 2 1 2 2 Two Equation System or = + Ε[ | ] , Ε[ | ] = E = +  = + ÷ = +    =     y y X 0 β y y 0 X β y 0 X X X 0 ε ε ε ε ε ε ε ε ε ε εε ε ε ε ε 11 12 12 22 2 = σ σ = σ σ σ I I I I
Part 15: Generalized Regression Applications OLS and GLS Each equation can be fit by OLS ignoring all others. Why do GLS?

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