The Banking Institute

Transcription

1 The Banking Institute The Lecturer: Kantorovich G.G. Classteacher: Demeshev B. Programme Econometrics I. An explanatory notes Purpose: The course will be a core one for the Banking Institute Master program Financial Analyst. The course is intended for studying during the first and the second semester of the Master level education. The course is a prerequisite for some both core and specialised courses of the curriculum. Requirements to students: The student should have knowledge and skills of Econometrics (Bachelor level), Time Series Analysis (Bachelor level), Microeconomics (intermediate level), Macroeconomics (intermediate level), and a number of mathematical and statistical courses such that Calculus of several variables, Linear algebra, Statistics, Computer sciences. Some necessary additional mathematical and statistical concepts and skills will be done within the course. The summary: The purpose of the course is not only to give students new skills in both econometric tools and their application to contemporary economic problems, especially in financial economics, but also to study theoretically econometric methods and to review some sections of econometrics on a solid theoretical background. The structure of the course includes strict derivation of basic properties of estimation methods but excludes proofs of the most analytically sophisticated results. The main studying purpose of such topics is to clear understanding of econometric ideas, assumptions under which econometric approaches can be applied. At the same time, the students should get skills in reading and understanding of the most advance econometric articles. 1

2 The course program provides lecturing and teaching classes, and regular self-study of students. Self-study includes deepening of theoretical material offered at lectures, and solutions of the offered home assignments. During each semester, one intermediate examination is set up. Educational task of the course: Because of study of material of the course, a student should master and be able to prove the basic facts of strict development of classical econometrics. She/he should also know main ideas of univariate and multivariable time-series analysis including Box-Jenkins approach, ARIMA (p,d,q) models, non-stationary time-series, unit root tests, cointegration, VAR and VECM. The student should have skills of application of the indicated tools and methods to researches in problems of Micro-, Macroeconomics and Finance. Forms of control: A current control of students' knowledge consists of appraisal students' activity during classes, marks for mid-term exams. The final-exam mark gives 70% of the final mark, 30% of the latter is given by the mid-term exam s mark. I. Teaching organisation and knowledge control Contact hours of the course are supposed to be equally shared between lectures and classes. All students will be supplied with the syllabi of the course, so it will allow practising lectures devoted to only main topics of its context and to use review lectures. Classes will be devoted to implementation of econometric methods given by classic scientific works periodicals. The specialised software R will be used during the classes. Each term will be finished with a written exam work. II. Hours distribution Classes Topic Lectures Subtotal Midterm exams Work without assistance Total hours 2

3 1 Introduction. General concept of regression. 2 The geometry of linear regression. 3 Classical linear regression (CLR). 4 OLS under assumption of normality. 5 Multicollinearity Models with stochastic regressors. 7 Maximum likelihood (ML) estimators and OLS-estimators under normality assumption. 8 Linear regression with heterogeneous observations. 9 Linear regression under non-spherical disturbances. GLS. 10 Linear regression under serial correlation. 11 Linear regression diagnostics. Specification errors. Model selection. 12 Dynamic regression with lagged variables. 13 Time series econometrics. 14 Panel Data Models. Fixed effects. Random effects Two-stage least squares. Three-stage least squares. 3

4 III. Structure and contents of the course 1. Introduction The nature of Econometrics and their place in the field of mathematical and economic knowledge. Implementation of econometric methods to analysis, modelling and forecasting of economic and social processes and events.. (1, , p. 1 8; 2, , p. 1 12; 3, , p. 1 21) 2. Concept of regression A hypothesis of existence of a relationship between economic variables. Explained and explanatory variables. Deviations from a relationship and an error term. Data generating process (DGP). Econometric model. Regression as a conditional expectation. Linear relationships. Estimation of the regression coefficients by Method of Moments (MM) under assumption of exogeneity. (1, , p ; 2, , p ; 3, 1.5, p ) 3. The geometry of linear regression Alternative criteria of goodness of fit of a linear relationship to the data. Interpolation as a linear problem and as a quadratic problem. Advantages and disadvantages of estimation of parameters of a linear regression line in a pure deterministic statement. Least square criterion and its implementation to linear regression line construction. Ordinary least square (OLS) in matrix notation. System of linear equations for finding OLS estimators for regression parameters. Properties of OLS estimators for regression parameters. Geometric interpretation of OLS for a regression with intercept. Orthogonal projectors and their properties. Symmetric idempotent matrices. Frisch- Waugh-Lovell theorem and its application to seasonality and detrending problens. Decomposition of the sum of squares of deviations of an explained 2 variable (TSS=ESS+RSS). Coefficient of determination R and its properties for regression with intercept. Regression through the origin. 4

5 Centered and uncentered coefficients of determination. Adjusted 2 coefficient of determination ( R ), its properties and its using for model analysis and selection. Orthogonal regression (both simple and multiple). The linear system for finding OLS-estimators in terms of sample variance-covariance matrix and sample correlation matrix. Invariance to shifts and scaling of explained and explanatory variables. ~ 2 Generalised coefficient of determination ( R ), suitable for both 2 regression with and without an intercept, ant its relations to R. Factor decomposition of coefficient of determination and its using in regression analysis. (1, , p ; 2, 4.3, p ; 2, , p ; 3, , p ) 4. Classical linear regression (CLR) Stochastic interpretation of deviations in a linear regression. Stochastic properties of the error term. OLS in CLR. Statistical properties of OLS-estimators for regression parameters and regression residuals. Gauss-Markov theorem and its interpretation for fixed regressors. Linear regression with linear parameter restrictions (Qa=q). Conditional OLS as an example of use of a-priori information Lagrangian function for conditional OLS problem. Conditional OLSestimators and their relationship with OLS-estimator. Unbiasedness of conditional OLS-estimators. Estimate of their variance-covariance matrix. A relationship between residual sums of squares for OLS and conditional OLS. (1, 6.6, p ; 2, 5.4, p ; 3, , p ) 5. OLS under assumption of normality Distributions of quadratic forms in components of a multiple normal vector. Statistical properties of OLS-estimators ( a ) and of a residual sum of 2 squares (RSS) (a) e, under assumption of normality. S k k Confidence intervals for regression parameters and a variance of an error term. Confidence regions for the vector of regression parameters and their linear combinations. Tests of general linear hypotheses. Using of a statistic 5

6 S( a( COLS)) S( a) m k. S( a) r Testing the significance of the complete regression (F-test). Confidence intervals for a value of explained variable y ( x 0 ) and its expectation. Forecast and post-prognosis (point and interval). Regression through the origin under assumption of normality. (1, , p ; 2, 5.4, p ; 3, , p ) 6. Multicollinearity Perfect and practical (imperfect) multicollinearity. Indicators of multicollinearity. Condition index of a matrix for determination of OLSestimates of the regression parameters: using of the observation matrix, of the sample variance-covariance matrix, of the sample correlation matrix. Principal component method and its implementation to estimation of linear regression parameters. Consequence of regressions with increasing number of orthogonal regressors and their representation in terms of original variables. Non-uniqueness of OLS-estimates under multicollinearity. Criteria for estimates selection. Regularisation of a problem of finding of OLSestimates. Biased estimation of regression parameters. Ridge regression. (1, 6.7, p ; 2, 6.5, p ) 7. Models with stochastic regressors Assumptions about random errors and stochastic regressors. Exogeneity and predeterminedness. Elements of Asymptotical Approach. Convergence in probability, convergence in mean square, convergence in distribution and their interrelation. Slutsky s theorem. Limiting distribution, limiting mean, limiting variance. Low of Large Numbers (LLN). Central Limit Theorems (CLT) in Lindberg-Levy and Lindberg Feller univariate and multivariate formulations. Gauss-Markov theorem and its interpretation for exogenous regressors. Gauss-Markov theorem and its interpretation for predetermined regressors. Models with stochastic regressors that are correlated with random errors. The instrumental variables (IV) method. Measurement errors. (1, 4.4, p ; 2, , p ; 3, , p ) 6

7 8. Maximum likelihood (ML) estimators and OLS-estimators under assumption of normality Maximum likelihood (ML) approach for estimation of parameters. Likelihood and loglikelihood functions. Properties of ML estimators. Cramer-Rao Minimal Variance Bound (MVB). Linear normal regression with independent and homoscedastic errors. Relationships between ML- and OLS- estimators of linear regression parameters under normality assumption. ML-estimators with known variance-covariance matrix of errors. Properties of the estimators. Regression equation when explanatory variables and the explained variable form a multiple normally distributed vector with non-singular variancecovariance matrix. Conditional expectation of the explained variable, its linearity with respect to the explanatory variables. Distribution of the explained variable under fixed values of the explanatory variables. ML-estimation of regression parameters under non-normal distribution of the errors. Classical asymptotic testing procedures: Likelihhod Ratio test (LR-test), Wald test, Lagrange Multiplier test (LM-test). (1, 4.4, p ; 1, 4.9, p ; 2, , p ; 3, , p ) 9. Linear regression with heterogeneous observations Properties of the observed objects and variables, which do require taking into consideration heterogeneity of the observations. A-priori division the observations into groups supposed homogeneous. Dummy variables, their setting and use. Interpretation of hypotheses about dummy variables. Dummy variables trap. Dummy variables for testing stability of regression coefficients (structural changes). Chow test and dummy-variable approach for comparing two regressions. (1, , p ; 2, 6.2, p ) 10. Linear regression under non-spherical disturbances Economic reasons for heteroscedasticity and its consequence for OLSestimators. The generalised least squares (GLS) estimators. Reduction of GLS to OLS with transformed data. Properties of the GLS estimators. 7

8 Heteroscedasticity. Weighted least squares and its reduction to OLS. Testing for heteroscedasticity. The Breusch-Pagan test. The Goldfeld-Quandt test. The Glejser test. The Park test. Spearman's rank correlation test for heteroscedasticity. Heteroscedasticity consistent standard errors (White standard errors). Feasible generalised least squares. Typical methods of parameterising of a variance-covariance matrix of the errors based on a-priori knowledge and results of testing for heteroscedasticity. (1, , p ; 2, , p ; 3, , p ) 11. Linear regression under serial correlation Economic nature of autocorrelation of errors. The first order autoregression scheme. Consequences of autocorrelation. The variance-covariance matrix of errors for autocorrelation, its inverse matrix. Detecting autocorrelation. Durbin-Watson statistic and its properties. Estimates for upper and lower bounds of the Durbin-Watson statistic's quantiles under uncorrelated normal disturbances in autoregression equation. Durbin- Watson test. Estimation of autocorrelation parameter based on Durbin-Watson statistic. Estimation of linear regression coefficients under autocorrelation. The Durbin two-step method. The Cochrane-Orcutt iterative procedure and its properties. GLS for autocorrelation with known value of autocorrelation parameter. Koyck transformation under known value of autocorrelation parameter and its relation to conditional OLS. Finding an autocorrelation parameter as an optimiser of a selected criterion for conditional OLS. Maximum likelihood estimation of linear regression coefficients under autocorrelation. Random initial value of the explained variable. AR(p) scheme for autocorrelated disturbances. ARMA scheme for autocorrelated disturbances. The Breusch-Godfrey (Lagrange multiplier) test. (1, , p ; 2, 8.5, p ; 3, , p ) 12. Linear regression diagnostics Influence of adding or removing of one observation on OLS estimates of regression parameters. Detecting influential observations. Detecting outliers by studentized residuals. Testing for normal errors. Grouping of observations due to results of regression diagnostics. 8

9 Recursive residuals, their properties and use for structural change analysis, testing for heteroscedasticity and testing for autocorrelation. The cusum test and the cusum-of-squares test. Influence of adding or removing of one explanatory variable on OLS estimates of regression parameters. (1, , p ; 2, 10.1, p ) 13. Specification errors Omission of relevant variables. Biasedness of estimates of regression coefficients and variance-covariance matrix. Inclusion of irrelevant variables. Unbiasedness of estimates of regression coefficients and variance-covariance matrix. Consequences of misspecification of regression under heteroscedasticity. Alternative functional forms of regression. Linear and log-linear regression. Box-Cox test and Zarembka scaling. (1, 8.4, p ; 2, 6.6, p ) 14. Model selection Different criteria of goodness of a model. Prior requirements to a "good" model. Sample subdivision into two subsamples: "learning" and "testing". Jacknife method. Choosing the best linear model given a set of explanatory variables. Method of all possible regressions, its advantages and disadvantages. Adjusted coefficient of determination as a criterion of the best model for a classic normal regression. Test of a hypothesis about inclusion a group of irrelevant variables, based on F-distribution. Correspondence between t and F values in multiple regression. (1, , p ) 15. Dynamic regression with lagged variables The reasons for lags in economics. Estimation of distributed-lag models. Ad hoc estimation of distributed-lag models (the Alt and Tinbergen approach). Geometric lags, the Koyck transformation. Autoregressive models with lagged explained variable. Long-run equilibrium value of the explained variable. Characteristic equation. Roots of the characteristic equation and stability. 9

10 Models of expectations. Naive model of expectations. The adaptive expectations model. Partial adjustment models. Rational expectations. The Almon polynomial lag. Rational lags. Estimation distributed-lag models. The Klein method. The instrumental variables (IV) method. Prediction with dynamic distributed-lag models. (1, , p ; 2, , p ; 3, , p ) 16. Time series econometrics Stochastic process and its main characteristics. Stationarity. Characteristics of stochastic processes (means, autocovariation and autocorrelation functions). Wold decomposition. Lag operator. Moving average models МА(q). Autoregressive models АR(р). Yull-Worker equations. Stationarity conditions. Autoregressive-moving average models ARMA (p,q). Coefficient estimation in ARMA (p,q) processes. Box- Jenkins approach. Information criteria. Non-stationary time series. ARIMA (p,d,q) models. The unit root problem. Spurious trends and regressions. Unit root tests (Dickey-Fuller). ADF test. KPSS and other unit root tests. Segmented trends and structure breaks. Time series co-integration. Co-integration regression. Testing of co-integration. Vector autoregression and co-integration. Cointegration and error correction model. VECM. Johansen test. Causality in time series models. Granger causality. (1, , p ; 2, 9.3, p ; 3, , p ) 17. Models with discrete dependent variables Qualitative variables and reasons for their use in econometric modelling. Concept of index function of explanatory variables. Stochastic models of discrete choice. Binary choice models. The linear probability model and the linear discriminant function. The logit and probit models. Testing of their properties. Multinomial choice models. Censored and truncated variables. The Tobit model. Duration (survival) models. Examples of qualitative dependent variables in microeconomics. (1, , p ; 2, 10.5, p ; 3, , p ) 10

11 18. Panel Data Models Fixed effects. The Within and Between estimators. Unbalanced panels and fixed effects. Random effects. Testing for random effects. Hausman s test. Unbalanced panels and random effects. Robast estimation of the fixed effects models. Heteroscedasticity in the random effects models. (1, , p ; 3, 12.6, p ; 3, 7.10, p ) 19. Simultaneous equations models Estimation of simultaneous equation models when errors are uncorrelated. Indirect least squares. Two-stage least squares. Three-stage least squares. The instrumental variables (IV) method. (1, , p ; 2, , p ; 3, , p ) VIII. List of reference Textbooks Principal manuals 1. Marno Verbeek. Guidebook into contemporary econometrics. М., «Научная книга», Magnus J. P., Katyshev P. K., Peresetsky A. A. Econometrics. An introductory course. Moscow, "Delo", (In Russian). 3. Dogherty C. Introduction in Econometrics. 5-th edition. Main books. 1. Greene W. H. Econometric Analysis. Fifth edition. Prentice Hall, Inc., Johnston J. Econometric Methods. Third edition. Mc Graw Hill Book Company, Davidson R., MacKinnon J. G. Econometric Theory and Methods. Oxford University Press, Johnston J. DiNardo J. Econometric Methods. Fourth edition. Mc Graw Hill Book Company,

12 5. Wooldridge J. M. Econometric Analysis of Cross Section and Panel Data. Cambridge, Mass., MIT Press, Ruud P. A. An Introduction to Classical Econometric Theory. New York, Oxford University Press, Maddala G. S. Introduction to Econometrics. Third edition. John Willey & Sons, Ltd., Pindyck R. S., Rubinfeld D. L. Econometric Models and Economic Forecasts. Third edition. Mc Graw Hill Book Company, Enders W. Applied Econometric Time Series. John Wiley & Sons, Inc., Mills, T.C. The Econometric Modelling of Financial Time Series. Cambridge University Press, Andrew C. Harvey. Time Series Models. Harvester wheatsheaf, Andrew С. Harvey. The Econometric Analysis of Time Series. Philip Allan, Kantorovich G. Lecture notes on Time Series Analysis. Economic Journal of Higher School of Economics. Moscow, 4, 2002, 1-4, (In Russian). Канторович Г.Г. Лекции по курсу «Анализ временных рядов». Экономический журнал ВШЭ, 4, 2002, 1-4, Banerjee, A., J.J. Dolado, and D.V. Hendry. Co-Integration, Error Correction, and Econometric Analysis of Non-Stationary Data. Oxford University Press, Maddala, G.S. and Kim In-Moo. Unit Roots, Cointegration, and Structural Change. Cambridge University Press, P. J. Brockwell, R. A. Davis, Introduction to Time Series and Forecasting. Springer, W. Charemza, D. Deadman. New Directions in Econometric Practice. Edward Elgar Publishing Limited, R. I. D. Harris. Using Cointegration Analysis in Econometric Modeling. Prentice Hall,