What is the MPC?

Learning Objectives 1.Use linear regression to establish the relationship between two variables 2.Show that the line is the line of best fit in precise sense 3.Show that the line links the conditional expectations of the variables 4.A more formal approach to hypothesis testing

Consumption Function Keynesian Consumption function  income today,  consumption today C=a+b*Y Econometrics : quantify economic relationships – What are “a” and “b”

Look at some data Look at individual level data: individual.dta Stata: scatter cons nmwage This gives a scatter plot with the first variable on the vertical axis and the second variable on the x axis

Look at data

Two Obvious facts 1.Observe many households at different income levels – There is clearly a positive relationship 2.cons depends on income but households with same income will not have same consumption – other factors influence consumption

How do we Calculate the MPC? Draw a line Many possible lines Intuition tells us that an “average” line would be a better estimate – We will show why this intuition is correct later Any line we draw (even the “best”) will not go through all the points – There will be deviations from the line

Conditional Expectation As an alternative to the line we could follow the logic of the gender example from the pervious section and look at conditional expectation Recall we answered the question of gender discrimination by comparing the average wage of two groups – The expected waged conditional on being a man or woman – we used the “summ if” command Formally – E(hwage|gender==1)= – E(hwage|gender==2)=

Conditional Expectation We can apply the same logic to the consumption function. Divide in two groups – Rich: nmwage>1000 – Poor: nmwage<1000 – generate rich=(nmwage>1000) Compare the average consumption of each using summ if

Conditional Expectation We get average consumption conditional on being rich or poor – E(Cons|Rich)= – E(Cons|Poor)= We can measure the marginal propensity of consume by taking the average income of each group – E(nmwage|Rich)= – E(nmwage|Poor)=

Conditional Expectation As you move from “poor” to “rich” your income rises by: – =661 – And consumption rises by: =490 So an estimate of the MPC would be 490/661 which is 0.74 This is a simple and intuitive method that builds on the logic of the gender example But…..

Obvious Problem The division between risk and poor was entirely arbitrary – Not natural like gender We throw away information by forcing individuals into one group or another Why not have 3 groups or any number of groups you like Intuitively the more the better – 10 group example But large numbers of groups would make calculations tedious and would always leave out some information

10 Income Groups

Compromise Imagine there are an infinity of groups but the conditional means are all related Specifically they have a linear relationship – E(cons|nmwage)=a+b*nmwage From now on we will write in more general notation – E(Y|X)=  1 +  2 X

Comment Note this is a restriction and it may not be true in the real world We impose it on the model – Looks reasonable in the consumption example If it isn't true then there might be a problem – Linear approx – GIGO Relationship doesn’t have to be linear but it does have to be parametric – We will see more on this later

So to Recap… We have data that appears to illustrate a relationship between two variables Intuitively we will put a line through the data that represents the data in some way What way? Two ways: 1.the line links all the conditional means 2.We choose the particular line that is closest to the data in a defined way These turn out to be the same

Draw a line to represent the data Show three data points for illustration

An Explanation Change in notation to be more general – Y is the LHS or dependent variable – X is the RHS or independent variable E(Y|X i ) = conditional mean i.e. does not describe every observation – Y i = E(Y|X i ) + u i – u i represents the deviation of each individual observation from the conditional mean Y i = E(Y|X i ) + u i Y i =  1+  2 X i + u i

What is U i ? Any factor other than income (X) which influences consumption (Y) – individual tastes and unpredictability approximation error because of assumption of linear relationship Later we will model this a random variable Perhaps with a normal distribution – Remember our warnings about the bell curve

OLS Estimation Find line of “best fit” Method of Ordinary Least Squares (OLS) to estimate  1  2 Objective: find estimates of  1  2 that minimizes the distance between the regression line and the actual data points, i.e. minimize the error terms Minimise the sum of squared deviations i.e. – Aside: why not absolute deviation or others?

Algebra of OLS min  i u i 2 i.e. min (u u 2 2 +u 3 2 +…+u i 2 ) Y i =  1 +  2 X i +u i => u i = Y i -  1 -  2 X  i u i 2 =  i (Y i -  1 -  2 X ) 2 = S(  1,  2 ) => sum of squared errors is a function of  1,  2 min S(  1,  2 ) = min  i (Y i -  1 -  2 X ) 2

To find minimum of any function: differentiate with respect to the arguments and set derivative = 0 i.e. find the point where the slope with respect to the argument = 0.

An Explanation b 1, b 2 are the Ordinary Least Squares (OLS) estimators of the true population parameters  1,  2. b 2 is the estimator of the slope coefficient: the slope coefficient measures the effect on y of a one unit change in x b 1 is the estimator of the intercept: the value of Y which occurs if X=0;

OLS in stata regress cons nmwage Source | SS df MS Number of obs = F( 1, 1328) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = cons | Coef. Std. Err. t P>|t| [95% Conf. Interval] nmwage | _cons | Estimated coef  i u i 2

The Answer The regression gives us a measure of the MPC The OLS estimate of the MPC is What use is this – Prediction – Causation – Statistical inference

Prediction We can use this to make predictions What would the consumption be if income were 2500 Cons= *2500 – This is equal to 1953 Be careful this is the predicted conditional mean – It is the next point on the line – What people with 2500 would consume on average – What they actually will consume is unknown because we don’t observe their U i

Predicted Consumption Predicted Cons Actual Consumption

Causation Remember all this only really identifies variables that move together It doesn’t show causation Need theory for that Obvious in the gender example (wages don’t cause changes in gender) Not obvious here causation can run both ways

Statistical Inference This estimate is generated from a sample Recall that the issue is whether we can use this fact about the sample to make statements about the world (“population”) The same issues of statistical inference arise in context of regression – OLS estimates are sample statistics just like the sample average wages in the gender example

More on the Residual (U i ) The residual is the difference between the line (conditional expectation) and the actual data Think of every individuals consumption as being made up of two bits – Conditional expectation – Residual The conditional expectation is that same for everyone with the same X (income) Residual is potentially different even for those with same income

Random Variable Residual is unknown in advance so we model it as a random variable Think of consumption being determined by systematic bit plus a roll of a dice See diagram – Actual consumption (expectation+residual) is distributed around the mean – All the means are linked

Each distribution is a slice in the data

Distribution of Y for two different “slices” of X

Empirical Distribution We can use the hist comand in stata to look at this Just as we got distribution of hwage for men and women hist cons, by(rich) norm We could do the same for any income group – hist cons if nmwage 900, norm All OLS does is draw a line through all the means Imagine laying all these distributions side by side

The “Slice” Around nmwage=1000

Distribution of Y f(Y|X) E(Y|X) X=600 X =900 X=1200

Putting it all together We usually assume that the residual is a normal random variable Seems reasonable in this case – But remember our concerns about normal So the full model is – Y i =  1 +  2 X i + u i – Where E(Y|X i )=  1 +  2 X i – And u i ~N(0,   