P1.T2. Stock & Watson Chapters 4 & 5. Bionic Turtle FRM Video Tutorials. By: David Harper CFA, FRM, CIPM
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1 P1.T2. Stock & Watson Chapters 4 & 5 Bionic Turtle FRM Video Tutorials By: David Harper CFA, FRM, CIPM Note: This tutorial is for paid members only. You know who you are. Anybody else is using an illegal copy and also violates GARP s ethical standards.
2 P1.T2. Stock & Watson, Chapters 4 & 5 Quantitative Agenda Introduction to Econometrics (Stock & Watson) Linear Regression with One Regressor (Ch. 4) Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals (Ch. 5)
3 P1.T2. Stock & Watson, Chapters 4 & 5 T2.Quantitative: Learning Spreadsheets Workbook T Single variable linear regression Exam Relevance (XLS not topic) Medium Worksheet Note: If you are unable to view the content within this document we recommend the following: MAC Users: The built-in pdf reader will not display our non-standard fonts. Please use adobe s pdf reader ( PC Users: We recommend you use the foxit pdf reader ( or adobe s pdf reader ( Mobile and Tablet users: We recommend you use the foxit pdf reader app or the adobe pdf reader app. All of these products are free. We apologize for any inconvenience. If you have any additional problems, please Suzanne at suzanne@bionicturtle.com.
4 Chapter 4: Linear Regression with One Regressor
5 P1.T2. Stock & Watson, Chapter 4: Linear Regression with One Regressor Explain how regression analysis in econometrics measures the relationship between dependent and independent variables. TestScore ClassSize other factors Y 0 ClassSize X u i 0 1 i i Dependent (regressand) Variable Independent (regressor) Variable 5
6 P1.T2. Stock & Watson, Chapter 4: Linear Regression with One Regressor Define and interpret a population regression function, regression coefficients, parameters, slope and the intercept. Slope coefficient Intercept coefficient error term Y i 0 1X i u i Parameters (regression coefficients) This error (u) in the PRF is estimated by the residual (e) in the SRF 6
7 P1.T2. Stock & Watson, Chapter 4: Linear Regression with One Regressor Define and interpret the stochastic error term (or noise component). The error term contains all the other factors aside from (X) that determine the value of the dependent variable (Y) for a specific observation. Y i 0 1X i u i 7
8 P1.T2. Stock & Watson, Chapter 4: Linear Regression with One Regressor Define and interpret a sample regression function, regression coefficients, parameters, slope and the intercept. One set of unknowable parameters (B1, B2). Each sample SRF Es mator (sta s c) Estimate stochastic PRF SRF Y B B X u i 0 1 i i ˆi Y b b X 0 1 i stochastic SRF Y b b X e i 0 1 i i 8
9 P1.T2. Stock & Watson, Chapter 4: Linear Regression with One Regressor Describe the key properties of a linear regression. Okay if non-linear in variables, but must be linear in parameters E( Y ) B B X 0 1 i 2 Linear variable, nonlinear parameter E( Y ) B B X i Nonlinear variable, Linear parameter 9
10 P1.T2. Stock & Watson, Chapter 4: Linear Regression with One Regressor Describe the method and assumptions of ordinary least squares for estimation of parameters: Estimate (conditional) mean of dependent variable The conditional distribution of u(i) given X(i) has a mean of zero Test hypotheses about nature of dependence To forecast the mean value of the dependent [X(i), Y(i)] are independent and identically distributed (i.i.d.) Correlation (dependence) is not causation Large outliers are unlikely 10
11 P1.T2. Stock & Watson, Chapter 4: Linear Regression with One Regressor Define and interpret the explained sum of squares, the total sum of squares, and the residual sum of squares Total (TSS) = Explained (ESS) + residual (SSR) n i 1 ˆ 2 i ESS Y Y SSR uˆ i 1 n i 1 2 i 2 TSS Y Y n i R 2 SSR SER n k 1 ESS SSR 1 TSS TSS 11
12 P1.T2. Stock & Watson, Chapter 4: Linear Regression with One Regressor Define and interpret the standard error of the regression (SER), and the regression R2. Total (TSS) = Explained (ESS) + residual (SSR) In the case of one regressor: In the general case of k = number of regressors (aka, independent variables): SER 2 e i SSR n 2 n 2 SER n SSR k 1 R 2 ESS 1 TSS SSR TSS 12
13 P1.T2. Stock & Watson, Chapter 4: Linear Regression with One Regressor Interpret the results of an ordinary least squares regression LINEST() Function Key for LINEST() B(1) B(0) B(1) B(0) slope intercept (0.48) (9.47) se (slope) se (intercept) R^2 se (y estimate) F df 7, ,315 ESS RSS TestScore STR (9.47) (0.48) Test Scores Test Scores versus Student-Teacher Ratio Student-teacher ratio 13
14 P1.T2. Stock & Watson, Chapter 4: Linear Regression with One Regressor Practice Question A five-year regression of monthly cotton price changes, such that the number of observations (n) equals 60, against average temperature changes produced a standard error of the regression (SER) of $1.20. If the total sum of squares (TSS) was $ dollars 2, what is the implied correlation coefficient? 14
15 P1.T2. Stock & Watson, Chapter 4: Linear Regression with One Regressor Practice Question A five-year regression of monthly cotton price changes, such that the number of observations (n) equals 60, against average temperature changes produced a standard error of the regression (SER) of $1.20. If the total sum of squares (TSS) was $ dollars 2, what is the implied correlation coefficient? As SER = SQRT[SSR/(n-df)], SSR = SER^2*(n-df). In this case (again, 2 coefficients = 2 df): SSR = 1.20^2*(60-2) = 83.52; R^2 = ESS/TSS = 1 - SSR/TSS = / = correlation = SQRT( ) =
16 Chapter 5: Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals
17 P1.T2. Stock & Watson, Chapter 7: Hypothesis Tests and Confidence Intervals in Multiple Regression Define, calculate, and interpret confidence intervals for regression coefficients. Limit = Coefficient ± [standard error critical c%] = Confidence Interval Coefficient SE Lower Upper Intercept Slope (B1)
18 P1.T2. Stock & Watson, Chapter 7: Hypothesis Tests and Confidence Intervals in Multiple Regression Define and interpret hypothesis tests about regression coefficients. TestScore STR (9.47) (0.48) STR: t statistic = ( )/0.48 = 4.75 p value 2 Tail ~ 0 % 18
19 P1.T2. Stock & Watson, Chapter 7: Hypothesis Tests and Confidence Intervals in Multiple Regression Define and differentiate between homoskedasticity and heteroskedasticity. The error term u(i) is homoskedastic if the variance of the conditional distribution of u(i) given X(i) is constant for i = 1,,n and in particular does not depend on X(i). Otherwise the error term is heteroskedastic. 19
20 P1.T2. Stock & Watson, Chapter 7: Hypothesis Tests and Confidence Intervals in Multiple Regression Describe the implications of homoskedasticity and heteroskedasticity. Implications of homoskedasticity: the OLS estimators remain unbiased and asymptotically normal. If heteroskedasticity, can use heteroskedastic-robust standard errors. 20
21 P1.T2. Stock & Watson, Chapter 5: Regression with a Single Regressor: Hypothesis Tests and Confidence Interva Explain the Gauss-Markov Theorem and its limitations, and alternatives to the OLS. The Gauss Markov theorem provides a theoretical justification for using OLS, but has two key limitations: 1. Its conditions might not hold in practice. In particular, if the error term is heteroskedastic as it often is in economic applications then the OLS estimator is no longer BLUE An alternative to OLS when there is heteroskedasticity of a known form, called the weighted least squares estimator. 2. Even if the conditions of the theorem hold, there are other candidate estimators that are not linear and conditionally unbiased; under some conditions, these other estimators are more efficient than OLS. 21
22 End of P1.T2. Stock & Watson, Chapters 4 & 5 Visit us on the
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