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1 Chap 14-1 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chapter 14 Additional Topics in Regression Analysis Statistics for Business and Economics 6 th Edition

2 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 14-2 Chapter Goals After completing this chapter, you should be able to: Explain regression model-building methodology Apply dummy variables for categorical variables with more than two categories Explain how dummy variables can be used in experimental design models Incorporate lagged values of the dependent variable is regressors Describe specification bias and multicollinearity Examine residuals for heteroscedasticity and autocorrelation

3 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 14-3 The Stages of Model Building Model Specification Coefficient Estimation Model Verification Interpretation and Inference Understand the problem to be studied Select dependent and independent variables Identify model form (linear, quadratic…) Determine required data for the study *

4 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 14-4 The Stages of Model Building Model Specification Coefficient Estimation Model Verification Interpretation and Inference Estimate the regression coefficients using the available data Form confidence intervals for the regression coefficients For prediction, goal is the smallest s e If estimating individual slope coefficients, examine model for multicollinearity and specification bias * (continued)

5 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 14-5 The Stages of Model Building Model Specification Coefficient Estimation Model Verification Interpretation and Inference Logically evaluate regression results in light of the model (i.e., are coefficient signs correct?) Are any coefficients biased or illogical? Evaluate regression assumptions (i.e., are residuals random and independent?) If any problems are suspected, return to model specification and adjust the model * (continued)

6 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 14-6 The Stages of Model Building Model Specification Coefficient Estimation Model Verification Interpretation and Inference Interpret the regression results in the setting and units of your study Form confidence intervals or test hypotheses about regression coefficients Use the model for forecasting or prediction * (continued)

7 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 14-7 Dummy Variable Models (More than 2 Levels) Dummy variables can be used in situations in which the categorical variable of interest has more than two categories Dummy variables can also be useful in experimental design Experimental design is used to identify possible causes of variation in the value of the dependent variable Y outcomes are measured at specific combinations of levels for treatment and blocking variables The goal is to determine how the different treatments influence the Y outcome

8 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 14-8 Dummy Variable Models (More than 2 Levels) Consider a categorical variable with K levels The number of dummy variables needed is one less than the number of levels, K – 1 Example: y = house price ; x 1 = square feet If style of the house is also thought to matter: Style = ranch, split level, condo Three levels, so two dummy variables are needed

9 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 14-9 Dummy Variable Models (More than 2 Levels) Example: Let condo be the default category, and let x 2 and x 3 be used for the other two categories: y = house price x 1 = square feet x 2 = 1 if ranch, 0 otherwise x 3 = 1 if split level, 0 otherwise The multiple regression equation is: (continued)

10 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Interpreting the Dummy Variable Coefficients (with 3 Levels) With the same square feet, a ranch will have an estimated average price of thousand dollars more than a condo With the same square feet, a split-level will have an estimated average price of thousand dollars more than a condo. Consider the regression equation: For a condo: x 2 = x 3 = 0 For a ranch: x 2 = 1; x 3 = 0 For a split level: x 2 = 0; x 3 = 1

11 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Experimental Design Consider an experiment in which four treatments will be used, and the outcome also depends on three environmental factors that cannot be controlled by the experimenter Let variable z 1 denote the treatment, where z 1 = 1, 2, 3, or 4. Let z 2 denote the environment factor (the blocking variable), where z 2 = 1, 2, or 3 To model the four treatments, three dummy variables are needed To model the three environmental factors, two dummy variables are needed

12 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Define five dummy variables, x 1, x 2, x 3, x 4, and x 5 Let treatment level 1 be the default (z 1 = 1) Define x 1 = 1 if z 1 = 2, x 1 = 0 otherwise Define x 2 = 1 if z 1 = 3, x 2 = 0 otherwise Define x 3 = 1 if z 1 = 4, x 3 = 0 otherwise Let environment level 1 be the default (z 2 = 1) Define x 4 = 1 if z 2 = 2, x 4 = 0 otherwise Define x 5 = 1 if z 2 = 3, x 5 = 0 otherwise Experimental Design (continued)

13 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap The dummy variable values can be summarized in a table: Z1Z1 X1X1 X2X2 X3X Z2Z2 X4X4 X5X Experimental Design: Dummy Variable Tables

14 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap The experimental design model can be estimated using the equation The estimated value for β 2, for example, shows the amount by which the y value for treatment 3 exceeds the value for treatment 1 Experimental Design Model

15 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Lagged Values of the Dependent Variable In time series models, data is collected over time (weekly, quarterly, etc…) The value of y in time period t is denoted y t The value of y t often depends on the value y t-1, as well as other independent variables x j : A lagged value of the dependent variable is included as an explanatory variable

16 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Interpreting Results in Lagged Models An increase of 1 unit in the independent variable x j in time period t (all other variables held fixed), will lead to an expected increase in the dependent variable of j in period t j in period (t+1) j 2 in period (t+2) j 3 in period (t+3) and so on The total expected increase over all current and future time periods is j /(1- ) The coefficients 0, 1,..., K, are estimated by least squares in the usual manner

17 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Confidence intervals and hypothesis tests for the regression coefficients are computed the same as in ordinary multiple regression ( When the regression equation contains lagged variables, these procedures are only approximately valid. The approximation quality improves as the number of sample observations increases.) Caution should be used when using confidence intervals and hypothesis tests with time series data There is a possibility that the equation errors i are no longer independent from one another. When errors are correlated the coefficient estimates are unbiased, but not efficient. Thus confidence intervals and hypothesis tests are no longer valid. Interpreting Results in Lagged Models (continued)

18 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Specification Bias Suppose an important independent variable z is omitted from a regression model If z is uncorrelated with all other included independent variables, the influence of z is left unexplained and is absorbed by the error term, ε But if there is any correlation between z and any of the included independent variables, some of the influence of z is captured in the coefficients of the included variables

19 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Specification Bias If some of the influence of omitted variable z is captured in the coefficients of the included independent variables, then those coefficients are biased… …and the usual inferential statements from hypothesis test or confidence intervals can be seriously misleading In addition the estimated model error will include the effect of the missing variable(s) and will be larger (continued)

20 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Multicollinearity Collinearity: High correlation exists among two or more independent variables This means the correlated variables contribute redundant information to the multiple regression model

21 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Multicollinearity Including two highly correlated explanatory variables can adversely affect the regression results No new information provided Can lead to unstable coefficients (large standard error and low t-values) Coefficient signs may not match prior expectations (continued)

22 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Some Indications of Strong Multicollinearity Incorrect signs on the coefficients Large change in the value of a previous coefficient when a new variable is added to the model A previously significant variable becomes insignificant when a new independent variable is added The estimate of the standard deviation of the model increases when a variable is added to the model

23 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Detecting Multicollinearity Examine the simple correlation matrix to determine if strong correlation exists between any of the model independent variables Multicollinearity may be present if the model appears to explain the dependent variable well (high F statistic and low s e ) but the individual coefficient t statistics are insignificant

24 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Assumptions of Regression Normality of Error Error values (ε) are normally distributed for any given value of X Homoscedasticity The probability distribution of the errors has constant variance Independence of Errors Error values are statistically independent

25 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Residual Analysis The residual for observation i, e i, is the difference between its observed and predicted value Check the assumptions of regression by examining the residuals Examine for linearity assumption Examine for constant variance for all levels of X (homoscedasticity) Evaluate normal distribution assumption Evaluate independence assumption Graphical Analysis of Residuals Can plot residuals vs. X

26 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Residual Analysis for Linearity Not Linear Linear x residuals x Y x Y x

27 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Residual Analysis for Homoscedasticity Non-constant variance Constant variance xx Y x x Y residuals

28 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Residual Analysis for Independence Not Independent Independent X X residuals X

29 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Excel Residual Output RESIDUAL OUTPUT Predicted House PriceResiduals Does not appear to violate any regression assumptions

30 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Heteroscedasticity Homoscedasticity The probability distribution of the errors has constant variance Heteroscedasticity The error terms do not all have the same variance The size of the error variances may depend on the size of the dependent variable value, for example When heteroscedasticity is present least squares is not the most efficient procedure to estimate regression coefficients The usual procedures for deriving confidence intervals and tests of hypotheses is not valid

31 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Tests for Heteroscedasticity To test the null hypothesis that the error terms, ε i, all have the same variance against the alternative that their variances depend on the expected values Estimate the simple regression Let R 2 be the coefficient of determination of this new regression The null hypothesis is rejected if nR 2 is greater than 2 1, where 2 1, is the critical value of the chi-square random variable with 1 degree of freedom and probability of error

32 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Autocorrelated Errors Independence of Errors Error values are statistically independent Autocorrelated Errors Residuals in one time period are related to residuals in another period Autocorrelation violates a least squares regression assumption Leads to s b estimates that are too small (i.e., biased) Thus t-values are too large and some variables may appear significant when they are not

33 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Autocorrelation Autocorrelation is correlation of the errors (residuals) over time Violates the regression assumption that residuals are random and independent Here, residuals show a cyclic pattern, not random

34 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap The Durbin-Watson Statistic The possible range is 0 d 4 d should be close to 2 if H 0 is true d less than 2 may signal positive autocorrelation, d greater than 2 may signal negative autocorrelation The Durbin-Watson statistic is used to test for autocorrelation H 0 : successive residuals are not correlated (i.e., Corr(ε t,ε t-1 ) = 0) H 1 : autocorrelation is present

35 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Testing for Positive Autocorrelation Decision rule: reject H 0 if d < d L H 0 : positive autocorrelation does not exist H 1 : positive autocorrelation is present 0dUdU 2dLdL Reject H 0 Do not reject H 0 Inconclusive Calculate the Durbin-Watson test statistic = d d can be approximated by d = 2(1 – r), where r is the sample correlation of successive errors Find the values d L and d U from the Durbin-Watson table (for sample size n and number of independent variables K)

36 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Negative Autocorrelation Decision rule for negative autocorrelation: reject H 0 if d > 4 – d L 0dUdU 2dLdL Reject H 0 Do not reject H 0 Inconclusive Negative autocorrelation exists if successive errors are negatively correlated This can occur if successive errors alternate in sign Reject H 0 Inconclusive 4 – d U 4 – d L 4

37 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Example with n = 25: Durbin-Watson Calculations Sum of Squared Difference of Residuals Sum of Squared Residuals Durbin-Watson Statistic Testing for Positive Autocorrelation (continued) Excel/PHStat output:

38 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Here, n = 25 and there is k = 1 one independent variable Using the Durbin-Watson table, d L = 1.29 and d U = 1.45 D = < d L = 1.29, so reject H 0 and conclude that significant positive autocorrelation exists Therefore the linear model is not the appropriate model to forecast sales Testing for Positive Autocorrelation (continued) Decision: reject H 0 since D = < d L 0 d U = d L =1.29 Reject H 0 Do not reject H 0 Inconclusive

39 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Dealing with Autocorrelation Suppose that we want to estimate the coefficients of the regression model where the error term ε t is autocorrelated Two steps: (i) Estimate the model by least squares, obtaining the Durbin-Watson statistic, d, and then estimate the autocorrelation parameter using

40 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Dealing with Autocorrelation (ii) Estimate by least squares a second regression with dependent variable (y t – ry t-1 ) independent variables (x 1t – rx 1,t-1 ), (x 2t – rx 2,t-1 ),..., (x k1t – rx k,t-1 ) The parameters 1, 2,..., k are estimated regression coefficients from the second model An estimate of 0 is obtained by dividing the estimated intercept for the second model by (1-r) Hypothesis tests and confidence intervals for the regression coefficients can be carried out using the output from the second model

41 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Chapter Summary Discussed regression model building Introduced dummy variables for more than two categories and for experimental design Used lagged values of the dependent variable as regressors Discussed specification bias and multicollinearity Described heteroscedasticity Defined autocorrelation and used the Durbin- Watson test to detect positive and negative autocorrelation

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