1. Tutorial 1: Misspecification
Matthew Robson
University of York
Econometrics 2

2. 𝐿 𝐶 𝑡 = log consumption, 𝐿 𝐼 𝑡 = log income, 𝐿 𝑊 𝑡 = log wealth
Assignment 5
Assume that the true model of consumers’ behaviour is:
𝐿 𝐶 𝑡 = 𝛽 0 + 𝛽 1 𝐿 𝐼 𝑡 + 𝛽 2 𝐿𝑊+ 𝑢 𝑡
Where:
𝐿 𝐶 𝑡 = log consumption, 𝐿 𝐼 𝑡 = log income, 𝐿 𝑊 𝑡 = log wealth
Estimate the simple log-linear consumption function:
𝐿 𝐶 𝑡 = 𝛽 0 ∗ + 𝛽 1 ∗ 𝐿 𝐼 𝑡 + 𝑢 𝑡 ∗
Over the period 1967q1-2002q4
(1)
(2)

3. Descriptive Statistics

4. OLS Results: True Model
𝐿 𝐶 𝑡 = 𝐿 𝐼 𝑡 𝐿𝑊+ 𝑢 𝑡
𝐼𝑛𝑡𝑒𝑟𝑝𝑟𝑒𝑡𝑎𝑡𝑖𝑜𝑛: 𝑖𝑓 𝑤𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑥 𝑏𝑦 1% 𝑤𝑒 𝑒𝑥𝑝𝑒𝑐𝑡 𝑦 𝑡𝑜 𝑐ℎ𝑎𝑛𝑔𝑒 𝑏𝑦 𝛽%.
𝑇ℎ𝑒 𝑝𝑎𝑟𝑡𝑖𝑎𝑙 𝑒𝑙𝑎𝑠𝑡𝑖𝑐𝑖𝑡𝑦, 𝑎𝑙𝑙 𝑒𝑙𝑠𝑒 ℎ𝑒𝑙𝑑 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡.

5. Fitted Model

6. OLS Results: Simple Model
𝐿 𝐶 𝑡 = 𝐿 𝐼 𝑡 + 𝑢 𝑡

7. Question a)
Under what conditions will the OLS estimates of (2) be unbiased and consistent for the true parameters 𝛽 0 and 𝛽 1 ?
True: 𝑌 𝑡 = 𝛽 0 + 𝛽 1 𝑋 1𝑡 + 𝛽 2 𝑋 2𝑡 + 𝑢 𝑡
Estimated: 𝑌 𝑡 = 𝛽 0 ∗ + 𝛽 1 ∗ 𝑋 1𝑡 + 𝑢 𝑡 ∗
Where: 𝑢 𝑡 ∗ = 𝑢 𝑡 + 𝛽 2 𝑋 2𝑡
True Model Deviation from the mean:
𝑦 𝑡 = 𝛽 1 𝑥 1𝑡 + 𝛽 2 𝑥 2𝑡 + 𝑢 𝑡
Where: 𝑥 𝑖𝑡 = 𝑋 𝑖𝑡 − 𝑋 𝑖

8. Question a) 𝛽 1 ∗ = ∑ (𝑥 1𝑡 𝑦 𝑡 ) ∑ 𝑥 1𝑡 2 We know:
Covariance of x & y
𝛽 1 ∗ = ∑ (𝑥 1𝑡 𝑦 𝑡 ) ∑ 𝑥 1𝑡 2
We know:
𝑦 𝑡 = 𝛽 1 𝑥 1𝑡 + 𝛽 2 𝑥 2𝑡 + 𝑢 𝑡 ∴
𝛽 1 ∗ = ∑( 𝑥 1𝑡 𝛽 1 𝑥 1𝑡 + 𝛽 2 𝑥 2𝑡 + 𝑢 𝑡 ) ∑ 𝑥 1𝑡 2
𝛽 1 ∗ = 𝛽 1 ∑ 𝑥 1𝑡 𝑥 1𝑡 + 𝛽 2 ∑ 𝑥 1𝑡 𝑥 2𝑡 +∑( 𝑥 1𝑡 𝑢 𝑡 ) ∑ 𝑥 1𝑡 2
𝛽 1 ∗ = 𝛽 1 + 𝛽 2 ∑ 𝑥 1𝑡 𝑥 2𝑡 ∑ 𝑥 1𝑡 2 + ∑( 𝑥 1𝑡 𝑢 𝑡 ) ∑ 𝑥 1𝑡 2
Sample variance of x
Expectation = 0

9. Question a) 𝐸 𝛽 1 ∗ = 𝛽 1 + 𝛽 2 ∑ 𝑥 1𝑡 𝑥 2𝑡 ∑ 𝑥 1𝑡 2
𝐸 𝛽 1 ∗ = 𝛽 1 + 𝛽 2 ∑ 𝑥 1𝑡 𝑥 2𝑡 ∑ 𝑥 1𝑡 2
If 𝛽 2 ∑ 𝑥 1𝑡 𝑥 2𝑡 ∑ 𝑥 1𝑡 2 is +ve (-ve) bias is upwards (downwards)
∑ 𝑥 1𝑡 𝑥 2𝑡 ∑ 𝑥 1𝑡 2 = 𝜆 1
𝑋 2𝑡 = 𝜆 0 + 𝜆 1 𝑋 1𝑡 + 𝑣 𝑡
𝛽 1 ∗ = biased if 𝜆 1 ≠0 𝑎𝑛𝑑 𝛽 2 ≠0
If 𝜆 1 and 𝛽 2 have same (opposite) sign upwards (downwards) bias
Bias

10. Question a)
If model (2) was the true model 𝛽 0 ∗ and 𝛽 1 ∗ would be unbiased and consistent (i.e. 𝛽 2 =0) estimators of 𝛽 0 and 𝛽 1
If model (2) suffers from omitted variable bias (i.e. 𝛽 2 ≠0) the estimates of 𝛽 1 ∗ will be biased and inconsistent, for 𝛽 1 , if 𝑋 1𝑡 and 𝑋 2𝑡 are correlated. The estimator of 𝛽 0 ∗ will be biased and inconsistent, for 𝛽 0 , even if 𝑋 1𝑡 and 𝑋 2𝑡 are uncorrelated:
𝐸 𝛽 0 ∗ = 𝛽 0 + 𝛽 2 𝜆 0
𝜆 0 = 𝑋 2𝑡 − 𝜆 1 𝑋 1𝑡
The only way the intercept is unbiased is if 𝛽 2 =0 (i.e is the true model) or 𝜆 1 = 𝑋 2𝑡 =0 or 𝜆 0 =0

11. Question b) Estimate the following equation:
𝐿 𝑊 𝑡 = 𝜆 0 + 𝜆 1 𝐿 𝐼 𝑡 + 𝑣 𝑡
Over the full sample period 1967q1-2002q4
In which direction do you think the OLS estimate of 𝛽 1 ∗ from equation (2) is biased for 𝛽 1 ?
(3)

12. OLS Results: Equation (3)
𝑋 2𝑡 = 𝜆 0 + 𝜆 1 𝑋 1𝑡 + 𝑣 𝑡
𝐿 𝑊 𝑡 =− 𝐿 𝐼 𝑡 + 𝑣 𝑡
𝐸 𝛽 1 ∗ = 𝛽 1 + 𝛽 2 𝜆 1
𝜆 1 = +ve sign, and 𝛽 2 is expected to be +ve ∴ expect bias to be +ve
𝐸 𝛽 0 ∗ <𝐸 𝛽 0 , as 𝜆 0 =−𝑣𝑒

13. Question b) Does our prediction hold true?
𝐿 𝐶 𝑡 = 𝐿 𝐼 𝑡 𝐿𝑊+ 𝑢 𝑡
𝐿 𝐶 𝑡 = 𝐿 𝐼 𝑡 + 𝑢 𝑡
Yes: 0.974>0.943, 𝛽 1 ∗ > 𝛽 1
The omitted variable, wealth, likely biases the coefficient upwards.
Yes: <0.564, 𝛽 0 ∗ < 𝛽 0
The omitted variable, wealth, likely biases the intercept downwards.

14. Question c)
What can be said about the precision of your estimates of equation (2)?
In general we know:
The error variance, 𝜎 2 , is incorrectly estimated
The standard errors of the estimated coefficients 𝑣𝑎𝑟 𝛽 are biased
Conventional t and F tests are of limited value (i.e. invalid)

15. 𝐶𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡
Question c)
From (1) the variance of the error term is 𝜎 2 . We have an estimate of this which is 𝜎 2 , this is equal to ∑ 𝑒 𝑖 2 𝑛−𝑘 = RSS 𝑛−𝑘 , where 𝑒 𝑖 are the residuals from the model, n is the sample size, and k is the number of parameters. Estimates of 𝜎 2 are likely to be different between the true and estimated model. Why? Consider (2)…
𝑣𝑎𝑟 𝛽 1 = 𝜎 2 ∑ 𝑥 1𝑡 2 1− 𝑟 12 2
𝑣𝑎𝑟 𝛽 1 ∗ = 𝜎 2 ∑ 𝑥 1𝑡 2
The variance of 𝛽 1 ∗ is a biased estimate for the true model. But what if 𝑟 12 2 =0, will the variance be the same? No. As 𝜎 2 is likely to be incorrect as well..
In general if 𝑟 12 2 ≥0 we could expect 𝑣𝑎𝑟 𝛽 1 ∗ <𝑣𝑎𝑟 𝛽 1 but remember that 𝜎 ∗ 2 is unlikely to be equal to 𝜎 2 .
𝑃𝑒𝑎𝑟𝑠𝑜𝑛
𝐶𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡

16. Question d)
Explain how you would use a Ramsey RESET test to check whether your model is specified correctly? Calculate the RESET test for your estimated model (2) and discuss the outcome of this test
What other test might you use to investigate correct model specification? Briefly explain how you would use this test in the example above.

17. Ramsey RESET Test
Step 1: From the assumed model obtain the estimated 𝑌 𝑖 (old)
Step 2: Re-run the assumed model including the estimated 𝑌 𝑖 in some form as an additional regressor (new)
Step 3: Then use the F-test to find out whether the increase in 𝑅 2 , by using the model in Step 2, is statistically significant
𝐹= 𝑅 𝑛𝑒𝑤 2 − 𝑅 𝑜𝑙𝑑 2 𝑛𝑜. 𝑛𝑒𝑤 𝑟𝑒𝑔𝑟𝑒𝑠𝑠𝑜𝑟𝑠 − 𝑅 𝑛𝑒𝑤 𝑛−𝑛𝑜. 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟𝑠 𝑖𝑛 𝑛𝑒𝑤 𝑚𝑜𝑑𝑒𝑙
Step 4: If the computed F statistic is significant (for example at 5%) we can reject the null hypothesis that the model is correctly specified

18. RESET Test: Manually Old Model: 𝑌 𝑡 = 𝛽 0 + 𝛽 1 𝑋 1𝑡 + 𝑢 𝑡 New Model:
𝑌 𝑡 = 𝛽 0 + 𝛽 1 𝑋 1𝑡 + 𝑢 𝑡
New Model:
𝑌 𝑡 = 𝛽 0 ∗ + 𝛽 1 ∗ 𝑋 1𝑡 + 𝛽 2 ∗ 𝑌 2𝑡 2 + 𝛽 3 ∗ 𝑌 3𝑡 3 + 𝑢 𝑡 ∗
Where: 𝑌 𝑖𝑡 = 𝛽 0 + 𝛽 1 𝑋 𝑖𝑡
𝐻 0 :𝑚𝑜𝑑𝑒𝑙 𝑖𝑠 𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑙𝑦 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑒𝑑
𝐻 1 : 𝐻 0 𝑖𝑠 𝑓𝑎𝑙𝑠𝑒
𝐹= − − −4 = ~ 𝐹 2,140
𝐹 2, ≈3.07, 𝐹 2, ≈4.79, ∴𝐹> 𝐹 𝑐 𝑎𝑡 5% 𝑎𝑛𝑑 1%, 𝑤𝑒 𝑟𝑒𝑗𝑒𝑐𝑡 𝑡ℎ𝑒 𝑛𝑢𝑙𝑙

19. RESET Test: Manually (alternative)
Old Model:
𝑌 𝑡 = 𝛽 0 + 𝛽 1 𝑋 1𝑡 + 𝑢 𝑡
New (alternative) Model:
𝑌 𝑡 = 𝛽 0 ∗ + 𝛽 1 ∗ 𝑋 1𝑡 + 𝛽 2 ∗ 𝑌 2𝑡 2 + 𝑢 𝑡 ∗
Where: 𝑌 𝑖𝑡 = 𝛽 0 + 𝛽 1 𝑋 𝑖𝑡
𝐻 0 :𝑚𝑜𝑑𝑒𝑙 𝑖𝑠 𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑙𝑦 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑒𝑑
𝐻 1 : 𝐻 0 𝑖𝑠 𝑓𝑎𝑙𝑠𝑒
𝐹= − − −3 = ~ 𝐹 1,140
𝐹 1, ≈3.92, 𝐹 1, ≈6.85, ∴𝐹> 𝐹 𝑐 𝑎𝑡 5% 𝑎𝑛𝑑 1%, 𝑤𝑒 𝑟𝑒𝑗𝑒𝑐𝑡 𝑡ℎ𝑒 𝑛𝑢𝑙𝑙

20. RESET Test: PcGive
𝑀𝑜𝑑𝑒𝑙⇒𝑇𝑒𝑠𝑡⇒𝑇𝑒𝑠𝑡⇒𝑂𝐾⇒𝑅𝐸𝑆𝐸𝑇 𝑡𝑒𝑠𝑡 𝑎𝑛𝑑 𝑅𝐸𝑆𝐸𝑇23 𝑡𝑒𝑠𝑡 𝐹 𝐹

21. Lagrange Multiplier (LM) test
Step 1: Estimate the restricted regression by OLS and obtain the residuals
Step 2: If the unrestricted model is the correct one the residuals from Step 1 will be related to the extra variables in the unrestricted model
Step 3: Regress the residuals from Step 1 on all the variables. This is the auxiliary regression
Step 4: For large sample sizes, the sample size (n) times by the 𝑅 2 from the auxiliary regression follows the chi-square distribution with degrees of freedom equal to the number of restrictions imposed by the restricted model.
𝑛 𝑅 2 ~ 𝜒 2 (𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑟𝑒𝑠𝑡𝑟𝑖𝑐𝑡𝑖𝑜𝑛𝑠)
asy

22. Lagrange Multiplier (LM) test
Unrestricted Model: 𝐿 𝐶 𝑡 = 𝛽 0 + 𝛽 1 𝐿 𝐼 𝑡 + 𝛽 2 𝐿𝑊+ 𝑢 𝑡
Restricted Model: 𝐿 𝐶 𝑡 = 𝛽 0 ∗ + 𝛽 1 ∗ 𝐿 𝐼 𝑡 + 𝑢 𝑡 ∗
Auxiliary Regression: 𝑢 𝑡 ∗ = 𝛼 0 + 𝛼 1 𝐿 𝐼 𝑡 + 𝛼 2 𝐿𝑊+ 𝑣 𝑡
Use the 𝑅 2 from the Auxiliary Regression to calculate the test statistic.
𝑛 𝑅 2 = 144∗ =1.268~ 𝜒 2 (1)
𝜒 ≈3.841, 𝜒 ≈6.635
𝐿𝑀< 𝜒 ⇒Do not reject the Null

23. RESET vs LM
RESET:
We do not get information of how to improve the model
However, it is good as we do not need to know which X’s are the issue
LM:
Allows us to choose the ‘better’ model
Relies on asymptotic assumptions, so low n means it is an inappropriate test