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

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

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

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 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?

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 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?

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 for b (compare the two examples). it is the difference in probability i.e. the distance on the distribution

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

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