Assignment 3 - Applied Linear Regression

1) Application of heteroskedasticity tests on MINITAB: 

    a) Compute the test of White and explain your statistical decision. 

    b) Compute the Goldfeld - Quandt test and explain whether or not you reject the null hypothesis. (choose number of points to remove, c = 4) 

2) Application of autocorrelation tests on MINITAB: 

    a) Compute Durbin Watson test and the LM test for autocorrelation. 

    b) Apply Cochrane-Orcutt procedure (only 2 iterations) to estimate the Model. 

    c) Compute again the Durbin Watson test for the transformed model. Was the autocorrelation issue resolved?

Helpful links:

[1] http://www.michaeljgrogan.com/serial-correlation-and-the-cochrane-orcutt-remedy/

[2] https://www.youtube.com/watch?v=6kXcCfdwSKI

<img src="images/giphy.gif” alt="hi" class="inline"/>

Above: Before and after Cochrane-Orcutt remedy. LS fit (left), residuals (right)

HOMOSCEDASTICITY

Fitting the model: 

> lmtest::bptest(model1, studentize=TRUE)

Call:

lm(formula = y ~ x, data = data)

Residuals:

     Min       1Q   Median       3Q      Max 

-2.78843 -1.44242  0.04263  1.32336  2.53848 

Coefficients:

             Estimate Std. Error t value Pr(>|t|)    

(Intercept) -19.04257    1.02536  -18.57 3.85e-16 ***

x             0.39013    0.01196   32.63  < 2e-16 ***

---

Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.713 on 25 degrees of freedom

Multiple R-squared:  0.9771,    Adjusted R-squared:  0.9761 

F-statistic:  1065 on 1 and 25 DF,  p-value: < 2.2e-16

Test of White is type of B-P Test:

studentized Breusch-Pagan test

data:  model1

BP = 7.7221, df = 1, p-value = 0.005455

• Since the p-value is less than 0.05, we reject the null hypothesis of homoskedasticity
  

Goldfeld-Quandt test

> lmtest::gqtest(formula = y ~ x, fraction = 4, data = data)

    Goldfeld-Quandt test

data:  y ~ x

GQ = 5.7356, df1 = 10, df2 = 9, p-value = 0.007417

alternative hypothesis: variance increases from segment 1 to 2

• ...
  

AUTOCORRELATION

Durbin-Watson Test

> lmtest::dwtest(formula = model1, data = data)

    Durbin-Watson test

data:  model1

DW = 0.33192, p-value = 6.346e-10

alternative hypothesis: true autocorrelation is greater than 0

• We test for serial correlation using the Durbin-Watson test and find that the p-value is less than 0.05, indicating possible serial correlation that needs to be corrected.
  
• Cochrane-Orcutt remedy only works when data is an AR(1) stationary process
  
• "In other words, taking a first difference of the data results in a stationary process whereby the data has a constant mean, variance and autocorrelation." [1]
  

Cochrane-Orcutt

> coch = cochrane.orcutt( lm(y~x,data=data))

> summary(coch)

Call:

lm(formula = y ~ x, data = data)

              Estimate Std. Error t value  Pr(>|t|)    

(Intercept) -19.031124   2.688410  -7.079 2.562e-07 ***

x             0.385092   0.025953  14.838 1.375e-13 ***

---

Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.9323 on 24 degrees of freedom

Multiple R-squared:  0.9017 ,  Adjusted R-squared:  0.8976

F-statistic: 220.2 on 1 and 24 DF,  p-value: < 1.375e-13

Durbin-Watson statistic 

(original):    0.33192 , p-value: 6.346e-10 < 0.05

(transformed): 1.76582 , p-value: 2.078e-01 -> 0.2078 > 0.05

• The transformed Durbin-Watson statistic is greater than 0.05 which means that serial correlation in the model has been removed.