What is MPC?

Hypothesis testing

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

Statistical Inference

• Same issue as with gender example• We may have a weird (slightly) sample

that by pure chance gives us 0.756 when the true value is 0.8

• We want a formal procedure that will answer the question:– Is the difference between 0.756 and 0.8 large

enough to enable use to reject the idea that the true MPC is 0.8?

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

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).

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

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.

OLS as a Normal RV

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OLS is Normal• Usually assume the estimators have a

normal distribution• How can we say this?

1. 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

2. 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

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

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

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Using one of the key properties of the normal distribution

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?

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

Mechanism for Hypothesis Test

1. State the Hypothesis we want to testH0: bMPC = 0.8 H1: bMPC ≠ 0.8

2. Calculate the distribution of bOLS assuming that H0 is true.• Stata provides the stn. error of beta

3. Locate our estimate on the distribution4. What is the probability that our estimate

would have come from this distribution?5. Does this lead us to believe the null

hypothesis?

95% of distribution

Distribution of b under H0

• 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 0.756 if the true value is 0.8

What is the probability?

• P(bOLS<0.756| bMPC=0.8)=?

• P(z <(0.756-0.8)/0.0307205))• P(z< -1.43226836 )= 0.07603• 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?

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”

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

Another Example

Use the individual consumption data1. State the Hypothesis we want to test

H0: bMPC = 0.6 H1: bMPC ≠ 0.6

2. Calculate the distribution of bOLS assuming that H0 is true.

3. Find our estimate on the distribution4. What is the probability that our estimate

would have come from this distribution?5. Does this lead us to believe the null

hypothesis?

95% of distribution

• 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

0.756 if the true value is 0.6• Probability is much less than 5%

– P(bOLS>0.756| bMPC=0.6)=0.00001

• 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

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 forb(compare the two examples).

– it is the difference in probability – i.e. the distance on the distribution

What Have We Learned?

1. OLS estimates the line of best fit

2. This line can be interpreted as the conditional expectation

3. OLS estimators are random variables so we need to do statistical inference

4. What matters is not absolute the difference between estimated and hypothesised values but the probability of that difference

What is Missing?

1. We have only one X variable

2. Our hypothesis test procedure is a little cumbersome

3. We have only dealt with hypotheses that relate to one coefficient being equal to some number