Lecture 8. Using the CLR Model

Transcription

1 Lecture 8. Using the CLR Model Example of regression analysis. Relation between patent applications and R&D spending Variables PATENTS = No. of patents (in 1000) filed RDEXP = Expenditure on research&development (in billions of 199 $) The data are time series for (34 observations) 1

2 First step in analysis: descriptive statistics and graphs Sample mean, standard deviation etc. of variables With time-series data, time-series plot (scale!). Note units are chosen to make the range of the variables comparable Scatterplot of PATENTS against RDEXP

3 Date: 09/6/01 Time: 13:35 Sample: PATENTS RDEXP Mean Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque-Bera Probability Observations 34 34

4 PATENTS RDEXP

5 PATENTS vs. RDEXP PATENTS RDEXP

6 Second step: estimate the relation PATENTS = α + β RDEXP + u t (Note that I use subscript t to reflect the nature of the data) We assume that this inexact linear relation satisfies the assumption of the Classical Linear Regression (CLR) model Assumption 1: u t, t = 1, K, n are random variables with E ( ) = 0 u t Assumption : X t, t = 1, K, n are deterministic, i.e. non-random, constants. Assumption 3 (Homoskedasticity) All u t ' s have the same variance, i.e. for t = 1, K,n t t = E( ut ) Var ( u ) = σ t 3

7 Assumption 4 (No serial correlation) The random errors u t and u s are not correlated for all t s = 1, K, n If these assumptions hold then the best estimator for α, β is the OLS estimator. The output of a computer program that computes the OLS estimates and other quantities is reproduced next. The formulas that the computer uses are OLS estimates ˆβ = n i= 1 ( X n i i= 1 X )( Y ( X i i X ) Y ) αˆ = Y βˆ X 4

8 Standard errors of estimates and standard error of regression i= 1 i= 1 ( X i Std( α ˆ) = s n n n X i X ) Std(β ˆ) = n i= 1 s ( X i X ) s = 1 n n e i i= 1 t-statistic T = αˆ / Std( αˆ) T = βˆ / Std( βˆ ) α β 5

9 R R n i= 1 = n i= 1 ( Yˆ i ( Y i Yˆ ) Y ) 6

10 Dependent Variable: PATENTS Method: Least Squares Date: 09/6/01 Time: 11:7 Sample: Included observations: 34 Variable Coefficient Std. Error t-statistic Prob. C RDEXP R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

11 Computer output Estimates: Interpretation coefficient RDEXP: if RDEXP changes by 1 unit, i.e. 1 billion of 199 $, then PATENTS changes by.7919, i.e. by about 79. Estimates and standard errors: We use the standard error to find a 95% confidence interval for β. The formula for the bounds is βˆ ± c Std( βˆ) with c the value such that Pr( T > c) =. 05 where T has a t distribution with 34-=3 degrees of freedom (df). From table in front cover c Hence, the 95% interval: [. 6764,.9075] Note if we use c = (rule of thumb) or c = (standard normal), the result is close. 7

12 Test: We test H 0 : β = 0 against H 1 : β 0 (two-sided test). If H : β 0 is true, then T = βˆ / Std( βˆ ) 0 = has a t distribution with 34-=3 degrees of freedom (df). We reject if T β > c, with c such that Pr( T > c) =. 05 for t distribution with 3 df. Hence c If H 1 : β > 0, then c is such that Pr( T > c) =. 05 for t distribution with 3 df or c For both H 1 s we reject H 0. Alternative way to report result of test is p- value p-value= = Pr( T > t ) = Pr( T > ) = β β where T β has the distribution that holds if H 0 is true, i.e. t distribution with 3 df. If the p- value is less than.05, we reject at the 5% level. Note F-statistic=(t-statistic β ) β 8

13 Fit of linear relation: The R seems high, but as we shall see that does not imply that all is well. 9

14 Reporting regression results Always report Estimates of the regression coefficients Their standard errors (preferred to t- ratios. Why?) Estimate of σ or σ, i.e. s or s R (least useful, but most users want to know) If the number variables is small two options in reporting the results As equation PATENTS = RDEXP (6.36) (.057) s = R =

15 Or in a table OLS estimates (standard errors) regression of no. of patents (in 1000) on R&D expenditure (in billion 199 $); Constant (6.37) RDEXP.79 (.057) Std. Error regression R.86 How well does the model fit? With time-series we can plot the observed Y t and fitted/predicted Yˆ t = αˆ + βˆ X t and also e = Y Yˆ (see graph) t t t The fit seems to be good, but there is a clear pattern in the OLS residuals. 11

16 Residual Actual Fitted

17 In the scatterplot I plot e t against e t 1, i.e. I investigate whether subsequent OLS residuals are correlated. Note the clear relation that indicates that Assumption 4 (no serial correlation) may not be correct. To check assumption 3 I plot et against RDEXP t. There is evidence of heteroskedasticity, i.e. the variance of u t (estimated by e t ) is related to X t and not the same for all t. To check the normal distribution of u t I plot the distribution of e t. 1

18 RESID vs. RESIDL RESID RESIDL

19 RESID vs. RDEXP RESID RDEXP

20 Kernel Density (Epanechnikov, h = 9.4) RESID

21 Forecasting If 170 RDEXP 1994 = (not in sample), then we predict PATENTSF 1994 = = 169. The variance of the prediction error is 1 ( X n+ 1 X ) sn+ 1 = s n n ( X i X ) i= 1 The square root (standard deviation of prediction error) is (compare with s = 11.17) A 95% prediction interval is, using equal to Y ˆ n sn+ 1 < Yn+ 1 < Yn < PATENTS 1994 < ˆ s n+ 1 13

22 Now a rather curious relation Births= No. of births (in 10000), West- Germany Storks=No. of stork couples in Baden-Wurtemberg Time-series

23 BIRTHS STORKS

24 BIRTHS vs. STORKS BIRTHS STORKS

25 Dependent Variable: BIRTHS Method: Least Squares Date: 09/6/01 Time: 3:43 Sample: Included observations: 1 Variable Coefficient Std. Error t-statistic Prob. C STORKS R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

26 Residual Actual Fitted

27 RESID vs. RESIDL 5 0 RESID RESIDL

28 Dependent Variable: DBIRTHS Method: Least Squares Date: 09/6/01 Time: 3:51 Sample(adjusted): Included observations: 11 after adjusting endpoints Variable Coefficient Std. Error t-statistic Prob. C DSTORKS R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

29 RESID vs. RESIDDIFL RESID RESIDDIFL