Panel Random-Coefficient Model (xtrc) Contents 1.Model & Methodology 2.Literature Using This Model 3.STATA Manual 4.Estimation Example and Simulation 1. Model & Methodology What is a random-coefficient model? Fixed(constant)-coeff. Model : Here, (i)The marginal effect of x on y is assumed to be constant over all i. (ii)The effect of unobservable factors is captured by the error-term. 1. Model & Methodology We, instead, may want to assume that the parameters of the equation is not constant. Random Coeff. Model : Some structures should be imposed on the parameters, since otherwise this is not estimable (and is also meaningless). Based on the assumptions on the structure, this model can be further classified. 1. Model & Methodology The Usual assumption is that,, where are independent of x. Then the main goal is to estimate and then to predict the values for each i using this. cf. Under the above assumption, estimating is not different from estimating a fixed-coeff. model with hetero-skedasticity. 1. Model & Methodology Random Coefficient models with panel data Why we use R.C. model of panel data? (Conventional models) does not allow the interaction of the individual specific and/or time varying differences with the included explanatory variables. (Hsiao and Pesaran (2004)) The general form of the model is, : Again, we need to assume some structures of 1. Model & Methodology Hsiao(1974) Model : where, (i) (ii) (iii) and 1. Model & Methodology Swamy(1970) Model : where, (i) (ii) (iii) This is what STATA runs with xtrc command. 1. Model & Methodology But in this case, why do not we estimate separate equations for each i ? According to Hsiao et al. (1989), (i)By reducing the # of parameters to be estimated, this improves efficiency. (ii)It reduces the multicollinearity problem due to the co-movement over time among the explanatory variables, by appealing to the panel-specific differences. 1. Model & Methodology In addition, when the assumption that the individual differences( ) are randomly distributed is true, there is no aggregation bias. 1. Model & Methodology Estimation method of the Swamy(1970) model : Then, Stacking the equations, we have where is a block diagonal matrix with along the main diagonal. 1. Model & Methodology GLS estimator of : where,, and Here we can see that is a matrix-weighted average of the panel-specific OLS estimator( ). (The weight is inversely proportional to their covariant matrix.) 1. Model & Methodology To estimate the above, we have to know. For this, we use two-step approach, following Swamy(1970). Here, OLS Panel specific estimator, Finally, where 1. Model & Methodology Prediction of (Swamy and Mehta(1975)) : : This is reported by the option, betas Test of parameter constancy (Swamy(1970)) H0: Test statistic: where 2. Literature Using This Model Swamy(1970) Revisits Grunfeld(1958)s firm investment function where is gross invest., is value of share and is capital stock. Test of parameter constancy: rejected. Estimation result: We will return to this example in section 4. 2. Literature Using This Model Hendricks et al. (1979) Goal: to estimate the level and shape of the electricity demand function. Data: control/test individuals during Connecticut Peak Load pricing experiment. (for 3 months) Individual demand function (daily/periodical) : where includes time-variable (knot variable) and dummy for the experiment group. 2. Literature Using This Model As a result, we get an estimate of each consumers demand function 2. Literature Using This Model But what determines ? This paper assumes that is a function of exogenous variables related to each j. where Z includes ownership of some appliances, # of people, climate factors, etc. In general, Z explains very well! With this estimation result, we can reconstruct the demand function for each profile-group. 2. Literature Using This Model Hsiao et al. (1989) Goal: to estimate the regional electricity demand in the state of Ontario Model: log-adjustment structure where x is the set of economic factors, w of climate factors and z of regional & seasonal specific factors. (the unit is municipality) 2. Literature Using This Model The parameter constancy test rejects H0. Using pure random coeff. model: cannot represent the difference in parameters which results from the region-specific factors. Hence this paper uses a mixed fixed-random coeff. model (assuming that the coeff. of z are fixed and others are randomly distributed). Using the root mean square error of the predicted values, we can show that this model performs well. 2. Literature Using This Model It is interesting to see that the mixed model outperforms the region-specific estimations in predicting regional demand behaviors. Also note that the pure random-coeff. model is the worst in prediction. Conclusion: Neither should we pool the data w/o taking account of heterogeneity across units, nor should we simply treat all regional heterogeneity as random draws from a common population. 3. STATA Manual 4. Estimation Example and Simulation Estimation example: revisit Grunfeld(1958) Data: invest2 data (invest. data for 5 firms) 4. Estimation Example and Simulation j Seemingly Unrelated Regression (for comparison) 4. Estimation Example and Simulation d Simulation N=5, T=20, Iteration=1000 Generation: where and We will do: (i)xtrc regress (ii) prediction for each (iii) Pooled OLS (iv) parameter constancy test (reject %) (v) calculate (vi) SUR for comparison 4. Estimation Example and Simulation Result 4. Estimation Example and Simulation Result 4. Estimation Example and Simulation Result 4. Estimation Example and Simulation Result 5. References Grunfeld, Y. (1958), The Determinants of Corporate Investment Unpublished Ph.D. Thesis, University of Chicago. Grunfeld, Y. and Z. Griliches (1960), Is Aggregation Necessarily Bad? The Review of Economics and Statistics Vol. 42 No.1 Hendricks, W., R. Koenker and D. Poirier (1979), Residentia Demand for Electricity Journal of Econometrics (9) Hsiao, C., D. Mountain, M. Chan and K. Tsui (1989), Modeling Ontario Regional Electricity System Demand Using a Mixed Fixed and Random Coefficients Approach Regional Science and Urban Economics (19) Hsiao, C. and M, Pesaran (2004), Random Coefficient Panel Data Models CESifo Working Paper No Swamy, P. (1970), Efficient Inference in a Random Coefficient Regression Model Econometrica Vol.38, No.2 Swamy, P. and J. Mehta (1975), Bayesian and Non-Bayesian Analysis of Switching Regressions and of Random Coefficient Regression Models Journal of American Statistical Association (70) Wooldridge, J. (2006) Introductory Econometrics