What is the MPC?

1. What is the MPC?

2. Learning Objectives • Use linear regression to establish the relationship between two variables • A formal procedure for statistical inference

3. Consumption Function • Keynesian Consumption function •  income today,  consumption today • C=a+b*Y • Econometrics : quantifyeconomic relationships • What are “a” and “b”

4. Look at data

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

6. How do we Calculate the MPC? • Draw a line • 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

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

10. 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|Xi) = conditional mean i.e. does not describe every observation • Yi = E(Y|Xi) +ui • uirepresents the deviation of each individual observation from the conditional mean • Yi= E(Y|Xi) + ui • Yi = 1+2 Xi + ui

11. What is Ui? • 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

12. 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 • Minimisethe sum of squared deviations i.e. • Aside: why not absolute deviation or others?

13. Algebra of OLS • min i ui2 i.e. min (u12 + u22+u32+…+ui2) • Yi = 1+2Xi+ui => ui = Yi - 1+2X • i ui2 = i (Yi - 1+2X)2 = S(1 , 2) • => sum of squared errors is a function of 1 , 2 • min S(1 , 2) = min i (Yi - 1+2X)2

14. 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.

16. An Explanation • b1, b2 are the ordinary least squares estimators of the true population parameters 1 , 2. • b2 is the estimator of the slope coefficient: the slope coefficient measures the effect on y of a one unit change in x • b1 is the estimator of the intercept: the value of Y which occurs if X=0; • We can use b1, b2 to make predictions about the value of Y for any given X.

17. Steps in the Analysis • Economic Model: state theory or hypothesis e.g. Keynesian model • Specify a mathematical model: single equation or several. • e.g. C=a+b*Y • Note: 1 & 2 from your other courses • Specify statistical model: how deal with “errors” caused by sampling • This is what makes statistics • Get data: I provide for this course • But for your own project you will need to get data

18. Steps in the Analysis • Estimate the parameters of the model that best fit to the data. e.g. what “b” gives the best fit • Reject the Model? • Test hypotheses regarding the parameters. e.g. is “b” = 0.8? • Not trivial because of sample • Prediction: “What if” • implicit in everything