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Basic Financial Econometrics Basic Financial Econometrics Alois Geyer Vienna University of Economics and Business [email protected] http://www.wu.ac.at/~geyer this version: June 24, 2021 preliminary and incomplete c Alois Geyer 2021 { Some rights reserved. This document is subject to the following Creative-Commons-License: http://creativecommons.org/licenses/by-nc-nd/2.0/at/deed.en US Contents 1 Financial Regression Analysis1 1.1 Regression analysis................................1 1.1.1 Least squares estimation.........................2 1.1.2 Implications................................3 1.1.3 Interpretation...............................4 1.2 Finite sample properties of least squares estimates..............6 1.2.1 Assumptions...............................8 1.2.2 Properties................................. 11 1.2.3 Testing hypothesis............................ 13 1.2.4 Example 6: CAPM, beta-factors and multi-factor models...... 15 1.2.5 Example 7: Interest rate parity..................... 19 1.2.6 Prediction................................. 21 1.3 Large sample properties of least squares estimates.............. 22 1.3.1 Consistency................................ 23 1.3.2 Asymptotic normality.......................... 25 1.3.3 Time series data............................. 26 1.4 Maximum likelihood estimation......................... 28 1.5 LM, LR and Wald tests............................. 31 1.6 Specifications................................... 33 1.6.1 Log and other transformations..................... 33 1.6.2 Dummy variables............................. 34 1.6.3 Interactions................................ 35 1.6.4 Difference-in-differences......................... 36 1.6.5 Example 11: Hedonic price functions.................. 37 1.6.6 Example 12: House price changes induced by siting decisions.... 38 1.6.7 Omitted and irrelevant regressors.................... 39 1.6.8 Selection of regressors.......................... 41 1.7 Regression diagnostics.............................. 43 1.7.1 Non-normality.............................. 43 1.7.2 Heteroscedasticity............................ 44 1.7.3 Autocorrelation.............................. 46 1.8 Generalized least squares............................ 51 1.8.1 Heteroscedasticity............................ 51 1.8.2 Autocorrelation.............................. 52 1.8.3 Example 19: Long-horizon return regressions............. 55 1.9 Endogeneity and instrumental variable estimation............... 57 1.9.1 Endogeneity................................ 57 1.9.2 Instrumental variable estimation.................... 59 1.9.3 Selection of instruments and tests................... 62 1.9.4 Example 21: Consumption based asset pricing............ 65 1.10 Generalized method of moments........................ 69 1.10.1 OLS, IV and GMM........................... 71 1.10.2 Asset pricing and GMM......................... 72 1.10.3 Estimation and inference........................ 74 1.10.4 Example 24: Models for the short-term interest rate......... 77 1.11 Models with binary dependent variables.................... 78 1.12 Sample selection................................. 82 1.13 Duration models................................. 84 2 Time Series Analysis 87 2.1 Financial time series............................... 87 2.1.1 Descriptive statistics of returns..................... 88 2.1.2 Return distributions........................... 91 2.1.3 Abnormal returns and event studies.................. 94 2.1.4 Autocorrelation analysis of financial returns.............. 97 2.1.5 Stochastic process terminology..................... 100 2.2 ARMA models.................................. 101 2.2.1 AR models................................ 101 2.2.2 MA models................................ 104 2.2.3 ARMA models.............................. 105 2.2.4 Estimating ARMA models........................ 106 2.2.5 Diagnostic checking of ARMA models................. 107 2.2.6 Example 35: ARMA models for FTSE and AMEX returns..... 108 2.2.7 Forecasting with ARMA models.................... 110 2.2.8 Properties of ARMA forecast errors.................. 112 2.3 Non-stationary models.............................. 115 2.3.1 Random-walk and ARIMA models................... 115 2.3.2 Forecasting prices from returns..................... 118 2.3.3 Unit-root tests.............................. 119 2.4 Diffusion models in discrete time........................ 124 2.4.1 Discrete time approximation...................... 126 2.4.2 Estimating parameters.......................... 126 2.4.3 Probability statements about future prices............... 129 2.5 GARCH models.................................. 131 2.5.1 Estimating and diagnostic checking of GARCH models........ 133 2.5.2 Example 49: ARMA-GARCH models for IBM and FTSE returns.. 133 2.5.3 Forecasting with GARCH models.................... 135 2.5.4 Special GARCH models......................... 136 3 Vector time series models 138 3.1 Vector-autoregressive models.......................... 138 3.1.1 Formulation of VAR models....................... 138 3.1.2 Estimating and forecasting VAR models................ 140 3.2 Cointegration and error correction models................... 143 3.2.1 Cointegration............................... 143 3.2.2 Error correction model.......................... 143 3.2.3 Example 53: The expectation hypothesis of the term structure... 145 3.2.4 The Engle-Granger procedure...................... 146 3.2.5 The Johansen procedure......................... 150 3.2.6 Cointegration among more than two series............... 155 3.3 State space modeling and the Kalman filter.................. 157 3.3.1 The state space formulation....................... 157 3.3.2 The Kalman filter............................ 158 3.3.3 Example 60: The Cox-Ingersoll-Ross model of the term structure.. 159 Bibliography 162 I am grateful to many PhD students of the VGSF program, as well as doctoral and master students at WU for valuable comments which have helped to improve these lecture notes. 1.1 Regression analysis 1 1 Financial Regression Analysis 1.1 Regression analysis We start by reviewing key aspects of regression analysis. Its purpose is to relate a depen- dent variable y to one or more variables X which are assumed to affect y. The relation is specified in terms of a systematic part which determines the expected value of y and a random part . For example, the systematic part could be a (theoretically derived) valuation relationship. The random part represents unsystematic deviations between ob- servations and expectations (e.g. deviations from equilibrium). The relation between y and X depends on unknown parameters β which are used in the function that relates X to the expectation of y. Assumption AL (linearity): We consider the linear regression equation y = Xβ + : 0 y is the n×1 vector (y1; : : : ; yn) of observations of the dependent (or endogenous) vari- able, is the vector of errors (also called residuals, disturbances, innovations or shocks), β is the K×1 vector of parameters, and the n×K matrix X of regressors (also called explanatory variables or covariates) is defined as follows: 0 1 1 x11 x12 ··· x1k B C B 1 x21 x22 ··· x2k C X = B . C : B . .. C @ . A 1 xn1 xn2 ··· xnk 0 k is the number of regressors and K=k+1 is the dimension of β=(β0; β1; : : : ; βk) , where β0 is the constant term or intercept. A single row i of X will be denoted by the K×1 column vector xi. For a single observation the model equation is written as 0 yi = xiβ + i (i = 1; : : : ; n): We will frequently (mainly in the context of model specification and interpretation) use formulations like y = β0 + β1x1 + ··· + βkxk + , where the symbols y, xi and represent the variables in question. It is understood that such equations also hold for a single observation. 1.1 Regression analysis 2 1.1.1 Least squares estimation A main purpose of regression analysis is to draw conclusions about the population using a sample. The regression equation y=Xβ+ is assumed to hold in the population. The sample estimate of β is denoted by b and the estimate of by e. According to the least squares (LS) criterion, b should be chosen such that the sum of squared errors SSE is minimized n n X 2 X 0 2 0 SSE(b) = ei = (yi − xib) = (y − Xb) (y − Xb) −! min : i=1 i=1 A necessary condition for a minimum is derived from SSE(b) = y0y − 2b0X0y + b0X0Xb; and is given by @SSE(b) = 0 : −2X0y + 2X0Xb = 0: @b Assumption AR (rank): We assume that X has full rank equal to K (i.e. the columns of X are linearly independent). If X has full rank, X0X is positive definite and the ordinary least squares (OLS) estimates b are given by b = (X0X)−1X0y: (1) The solution is a minimum since @2SSE(b) = 2X0X @b2 is positive definite by assumption AR. It is useful to express X0y and X0X in terms of the sums n n 0 X 0 X 0 X y = xiyi X X = xixi i=1 i=1 to point out that the estimate is related to the covariance between the dependent variable and the regressors, and the covariance among regressors. In the special case of the simple regression model y=b0+b1x+e with a single regressor the estimates b1 and b0 are given by syx sy b1 = 2 = ryx b0 =y ¯ − b1x;¯ sx sx where syx (ryx) is the sample covariance (correlation) between y and x, sy and sx are the sample standard deviations of y and x, andy ¯ andx ¯ are their sample means. 1.1 Regression analysis 3 1.1.2 Implications By the first order condition the OLS estimates satisfy the normal equation (X0X)b − X0y = −X0(y − Xb) = −X0e = 0; (2) which implies that each column of X is uncorrelated with (orthogonal to) e. If the first column of X is a column
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