1. What is MPC?
Hypothesis testing

2. Is the MPC 0.8? Typically we tell intro macro students that MPC is 0.8
Our OLS estimate is 0.756
Does this mean that 0.8 is wrong?
In this sample: yes!
But in the population: not clear

3. Statistical Inference
Same issue as with gender example
We may have a weird (slightly) sample that by pure chance gives us when the true value is 0.8
We want a formal procedure that will answer the question:
Is the difference between and 0.8 large enough to enable use to reject the idea that the true MPC is 0.8?

4. OLS Estimates are RV
The first step in a formal procedure is to note that the OLS estimators are themselves random variables.
The precise value of the estimate depends on precise sample
Since sample choice is random so is the precise value of the estimate
If we observed the entire population there would be no distribution

5. OLS Estimator is RV
The key issue is that both are functions of the data so the precise value of each estimate will depend on the particular data points included in the sample which is random
This observation is the basis
statistical inference
and all judgments regarding the quality of estimates (see later).

6. OLS as RV Sounds Weird? How can a random estimate be any use?
The estimate is not entirely random
It will be close to the true value on average (more later)
But deviation from truth is random

7. OLS as RV
To illustrate the impact of the sample on estimates, try different samples
Split the sample into 100 mini samples and run regression
Different samples will produce different betas
Plot histogram of answers
Is it “normal”?
In reality we don’t calculate the distribution using this method.

8. OLS as a Normal RV

9. OLS is Normal Usually assume the estimators have a normal distribution
How can we say this?
OLS estimators are linear functions of the data so if the data has a normal distribution then so will the OLS estimators
This makes use of one of the properties of Normal distributions that we reviewed
The data will have a normal distribution if the residual has a normal distribution
True because model is linear: Yi = 1+2Xi+ui
Use The CLT (more later) which applies to large samples

10. Why Normality? The assumption of normality is made for convenience
Easier to work with predefined distribution than histograms
Is it reasonable?
We had our doubts previously
Example: share prices
Econometricians must be honest with themselves

11. Distribution of bOLS “Sampling distribution of the estimator”
OLS is unbiased so the distribution is cantered on the true value
Its variance depends on the variance of the residuals
Estimate produced by stata
bOLS is normally distributed

12. Using one of the key properties of the normal distribution

13. Hypothesis Testing Non-trivial because of sampling distribution
Is the evidence consistent with the null hypothesis being true?
“Null hypothesis” is our preconceived notion
General approach
what would the distribution look like if the (null) hypothesis was true?
Where on the distribution is our estimate?
How likely is our estimate to occur when the hypothesis is true?
How likely is the hypothesis to be true?

14. A Criminal Trial Metaphor of a criminal trial
We have the hypothesis that the accused is innocent
We ask if the evidence is consistent with the hypothesis being true
If not we can reject the hypothesis
If yes we fail to reject
Note: don’t “accept” the hypothesis

15. Mechanism for Hypothesis Test
State the Hypothesis we want to test
H0: bMPC = 0.8 H1: bMPC ≠ 0.8
Calculate the distribution of bOLS assuming that H0 is true.
Stata provides the stn. error of beta
Locate our estimate on the distribution
What is the probability that our estimate would have come from this distribution?
Does this lead us to believe the null hypothesis?

16. Distribution of b under H0

17. Any estimate is possibly consistent with any hypothesis
Always the possibility of a real fluke even with no mistakes
But some are clearly more likely than others
We can measure the probability of an estimate occurring if we know the distribution of the estimator
Clearly it is possible to get an estimate of if the true value is 0.8

18. What is the probability?
P(bOLS<0.756| bMPC=0.8)=?
P(z <( )/ ))
P(z< )=
So there is an 8% chance that our estimate (or lower) would have arisen in a world where the true value is 0.8
How do we interpret this?

19. So what is the answer?
What does this say about the likelihood of 0.8 being the true value?
Free to make your own judgement
8% seems a little low
but 5% is the usual threshold
some times 10% or 1% is the threshold
So we cannot use this data to reject the idea that true value is 0.8
“Cannot reject the Null Hypothesis”

20. So what is the answer?
If we use 10% as our threshold then we can reject the hypothesis
Be sure you understand the differences between the two answers
The criminal trial metaphor may help
Now days it has become usual for researchers to report the probability

21. Another Example Use the individual consumption data
State the Hypothesis we want to test
H0: bMPC = 0.6 H1: bMPC ≠ 0.6
Calculate the distribution of bOLS assuming that H0 is true.
Find our estimate on the distribution
What is the probability that our estimate would have come from this distribution?
Does this lead us to believe the null hypothesis?

23. Again any estimate could be consistent with this hypothesis
What is the probability of a fluke now?
Clearly fairly unlikely to get an estimate of if the true value is 0.6
Probability is much less than 5%
P(bOLS>0.756| bMPC=0.6)=
So we “can reject” the hypothesis
i.e. We can use this data to reject the notion that the true value is 0.6
Same result for any threshold unless ridiculously small

24. Some Comments
Criminal Trial metaphor: The null hypothesis (innocence) will only be overturned if there is overwhelming evidence
What constitutes “overwhelming”
Not with regard to the size of the difference between values for b (compare the two examples).
it is the difference in probability
i.e. the distance on the distribution

25. What Have We Learned? OLS estimates the line of best fit
This line can be interpreted as the conditional expectation
OLS estimators are random variables so we need to do statistical inference
What matters is not absolute the difference between estimated and hypothesised values but the probability of that difference

26. What is Missing? We have only one X variable
Our hypothesis test procedure is a little cumbersome
We have only dealt with hypotheses that relate to one coefficient being equal to some number