1. Regression Method

2. Chapter Topics
Multiple regression
Autocorrelation

3. Regression Methods
To forecast an outcome (response variable, dependent variable) of a study based on a certain number of factors (explanatory variables, regressors).
The outcome has to be quantitative but the factors can either by quantitative or categorical.
Simple Regression deals with situations with one explanatory variable, whereas multiple regression tackles case with more than one regressors.

4. Simple Linear Regression
Collect data
Population
J $
Unknown Relationship
Random Sample

5. Multiple Regression Two or more explanatory variables
Multiple linear regression model
where  is the error term and  ~ N(0, 2)
Multiple Linear Regression Equation
Estimated Multiple Linear Regression Equation

6. Multiple Regression Least Squares Criterion
The formulae for the regression coefficients b0, b1, b2, bp involve the use of matrix algebra. We will rely on computer software packages to perform the calculations.
bi represents an estimate of the change in Y corresponding to a one-unit change in Xi when all other independent variables are held constant.

7. Multiple Regression R2=SSR/SST=1-SSE/SST Adjusted R2 ( )
where n is the number of observations
and p is the number of independent variables
The Adjusted R2 compensates for the number of independent variables in the model. It may rise or fall.
It will fall if the increase in R2 due to the inclusion of additional variables is not enough to offset the reduction in the degrees of freedom.

8. Test for Significance Test for Individual Significance: t test
Hypothesis
Test statistic
Decision rule: reject the null hypothesis at α level of significance if
, or
p-value < α

9. Test for Significance Testing for Overall Significance: F test
Test whether the multiple regression model as a whole is useful to explain Y, i.e., at least one X–variable in the regression model is useful to explain Y.
Hypothesis
H0 : all slope coefficients are equal to zero
(i.e. β1 = β2 =…= βp =0)
Ha : not all slope coefficients are equal to zero

10. Test for Significance Testing for Overall Significance: F test
Test statistic
Decision rule: reject null hypothesis if
F > Fα is based on an F distribution with p degrees of freedom in the numerator and n – p –1 degrees of freedom in the denominator, or
p-value < α

11. Example: District Sales
Use both target population and per capita discretionary income to forecast district sales.

12. Example: District Sales
Excel output

13. Example: District Sales
Multiple regression model
where
Y = district sales
X1 = target population
X2 = per capita discretionary income
Multiple Regression Equation
Using the assumption E( ) = 0, we obtain

14. Example: District Sales
Estimated Regression Equation
b0, b1, b2 are the least squares estimates of 0, 1, 2. Thus
For this example,
Predicted sales are expected to increase by gross when the target population increases by one thousand, holding per capita discretionary income constant.
Predicted sales are expected to increase by gross when per capita discretionary income increase by one dollar, holding population constant.

15. Example: District Sales
t Test for Significance of Individual Parameters
Hypothesis
Decision rule
For  = .05 and d.f. = 15 – 2 – 1 = 12, t.025 = 2.179
Reject H0 if |t| > 2.179
Test statistic
Conclusions
Reject H0: 1 = Reject H0: 2 = 0

16. Example: District Sales
To test whether sales are related to population and per capita discretionary income
Hypothesis
H0 : β1 = β2 =0
Ha : not both β1 and β2 equal to zero
Decision Rule
For  = .05 and d.f. = 2, 12: F.05 = 3.89
Reject H0 if F > 3.89.
Test statistic
F = MSR/MSE = 26922/4.74 =
Conclusion
Reject H0, sales are related to population and per capita discretionary income.

17. Example: District Sales
R2 = 99.89% means that 99.89% of total variation of sales can be explained by its linear relation with population and per capita discretionary income.
Ra2 = 99.88%. Both R2 and Ra2 mean the model fits the data very well.

18. Regression Diagnostics
Model assumptions about the error term 
The error  is a random variable with mean of zero, i.e., E() = 0
The variance of , denoted by  2, is the same for all values of the independent variable(s), i.e., Var() =  2
The values of  are independent.
The error  is a normally distributed random variable.

19. Regression Diagnostics
Residual analysis: validating model assumptions
Calculate the residuals and check the following.
Are the errors normally distributed?
Normal probability plot
Is the error variance constant?
Plot of residuals against
Are the errors uncorrelated (time series data)?
Plot of residuals against time periods
Are there observations that are inaccurately recorded or do not belong to the target population?
Double check the accuracy of outliers and influential observations.

20. Autocorrelation
Autocorrelation is present if the disturbance terms are correlated. Three issues need to be addressed.
How does autocorrelation arise?
How to detect autocorrelation?
Alternative estimation strategies under autocorrelation

21. Causes of Autocorrelation
Omitting relevant regressors
Suppose the true model is
But the model is mis-specified as
That is,
If X2t is correlated with X2,t-1, νt is also correlated with νt-1. This is particularly serious if X2t represents a lagged dependent variable.

22. Causes of Autocorrelation
Specification errors in the functional form
Suppose the true model is
But the model is mis-specified as
νt would tend to be positive
for X<A and X>B, and
negative for A<X<B.

23. Causes of Autocorrelation
Measurement errors in the variables
Suppose Yt = Yt* + νt
where Y is the observed value, Y* is the true value and ν is the measurement error. Hence, the true model is
and the observed model is
Given a “common” measurement method, it is likely that measurement errors in period t and t-1 are correlated.

24. Causes of Autocorrelation
Pattern of business cycle
Time-series data relating to business and economics often exhibit pattern of business cycle. Sluggishness during recession persists over a certain time period while prosperity in bloom continues for a certain duration of time. It is apparent that successive observations tend to be correlated.

25. Testing for First Order Autocorrelation
The error term in time period t is related to the error term in time period t–1 by the equation εt = ρεt-1 + at , where at ~ N(0, σa2).
Use Durbin-Watson test to test the existence of first order autocorrelation

26. Testing for First Order Autocorrelation
Durbin-Watson test
For positive autocorrelation
H0 : The error terms are not autocorrelated (ρ = 0)
Ha : The error terms are positively autocorrelated (ρ > 0)
For negative autocorrelation
Ha : The error terms are negatively autocorrelated (ρ < 0)
For positive or negative autocorrelation
Ha : The error terms are positively or negatively autocorrelated
(ρ  0)
Test statistic

27. Testing for First Order Autocorrelation
where r is the sample autocorrelation coefficient expressed as

28. Testing for First Order Autocorrelation
In “large samples”, DW  2(1–r)
If the disturbances are uncorrelated, then r = 0 and DW  2
If negative first order autocorrelation exists, then r<0 and DW > 2
If positive first order autocorrelation exists, then r>0 and DW < 2
Exact critical values of the Durbin-Watson test cannot be calculated. Instead, Durbin-Watson established upper (dU) and lower (dL) bounds for the critical values. They are for testing first order autocorrelation only.

29. Testing for First Order Autocorrelation
Test for positive autocorrelation
H0 : ρ = 0
Ha : ρ > 0
Decision rules
If DW < dL,α, we reject H0.
If DW > dU,α, we do not reject H0.
If dL,α ≤ DW ≤ dU,α, the test is inconclusive.

30. Example: Company Sales
The Blasidell Company wished to predict its sales by using industry sales as a predictor variable.

31. Example: Company Sales
From the scatter plot, a linear regression model is appropriate

32. Example: Company Sales
SAS output

33. Example: Company Sales
Estimated regression equation
The market research analyst was concerned with the possibility of positively correlated errors. Using the Durbin-Watson test:
H0 : ρ = 0
Ha : ρ > 0

34. Example: Company Sales
Suppose α = For n=20 (n denotes the number of observations) and k’ =1 (k’ denotes the number of independent variables),
dL = 0.95 and dU=1.15.
Since DW < dL, we conclude that the error terms are positively autocorrelated.

35. Testing for First Order Autocorrelation
Remark
In order to use the Durbin-Watson table, there must be an intercept term in the model.

36. Testing for First Order Autocorrelation
Test for negative autocorrelation
H0 : ρ = 0
Ha : ρ < 0
Decision rules
If 4 – DW < dL,α, we reject H0.
If 4 – DW > dU,α, we do not reject H0.
If dL,α ≤ 4 – DW ≤ dU,α, the test is inconclusive.

37. Testing for First Order Autocorrelation
Test for positive or negative autocorrelation
H0 : ρ = 0
Ha : ρ  0
Decision rules
If DW < dL,α/2 or 4 – DW < dL,α/2, we reject H0.
If DW > dU,α/2 and 4 – DW > dU,α/2 , we do not reject H0.
If dL,α/2 ≤ DW ≤ dU,α/2 or dL,α/2 ≤ 4 – DW ≤ dU,α/2 , the test is inconclusive.

38. Testing for First Order Autocorrelation
Remarks
The validity of the Durbin-Watson test depends on the assumption that the population of all possible residuals at any time t has a normal distribution.
Positive autocorrelation is found in practice more commonly than negative autocorrelation.
First-order autocorrelation is not the only type of autocorrelation.

39. Solutions to Autocorrelation (1)
1. Re-examine the model. The typical causes of autocorrelation are omitted regressors or wrong functional forms.
2. Go for alternative estimation strategy. Several approaches are commonly used. The approach considered here is the two-step Cochrane-Orcutt procedure.
Consider the following model with AR(1) disturbances :
(1)
with

40. Solutions to Autocorrelation (2)
Since equation (1) holds true for all observation, in terms of the (t-1)th observation, we have
(2)
where
Now, multiply (2) by r, we obtain
(3)
Subtracting (3) from (1), we get
That is,
(4)
Note that the ut’s are uncorrelated. However, r is unknown and needs to be estimated.

41. Two-step Cochrance-Orcutt
Estimate equation (1) by Least Squares method and obtain the
resulting residuals et’s. Regress et = ret-1 + ut and obtain
Substitute r into equation (4) and obtain OLS estimates of coefficients based on equation (4).

42. The following table represents the annual U.S. personal consumption
expenditure (C) in billions of 1978 dollars from 1976 to 1990 inclusively :

43. An OLS linear trend model has been fitted to the above data, and it gives
the following residuals :

44. To test for positive first order autocorrelation in the error and hence
estimate a model for this error process, consider
H0 : r = 0
Ha : r  0
Using the Durbin-Watson test,

45. When k’ =1 and n=15,
dl = 1.08, du = 1.36
Hence we reject H0
By regressing et on et-1, we obtain r = 0.79
Hence the error process is
Re-estimate the trend model for consumption using the two-step
Cochrane-Orcutt procedure.

46. Using the transformed model
with t=1 indicating year 1976, sequentially until t=15 representing year
1990, the transformed data are tabulated in following table.

47. Applying OLS to the transformed data yields
That is,
are parameter estimates of the original model.

48. Note that
1. Because lagged values of Y and X had to be formed, we are left with n-1 observations only
2. The estimate r is obtained based on OLS estimation assuming a standard linear regression model satisfying all classical assumptions. It may not be efficient estimator of r. This leads to the iterative Cochrane-Orcutt estimator.

49. Chapter Summary Simple linear regression Multiple regression
Regression on Dummy Variables
Autocorrelation
Durbin-Watson test
Two step Cochrane Orcutt procedure