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

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0 500 1000 1500 2000net monthly wage

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)=6.701875 – E(hwage|gender==2)= 5.451302

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)= 1024.11 – E(Cons|Poor)= 534.33

• We can measure the marginal propensity of consume by taking the average income of each group– E(nmwage|Rich)= 1282.42– E(nmwage|Poor)= 621.14

Conditional Expectation• As you move from “poor” to “rich” your

income rises by:– 1282-621=661– And consumption rises by: 1024-534=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 Groups0

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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)=b1+b2X

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

Y E(Y|X)=1+2X

Y1 u1

Y3 u3

Y2 u2

1 X2 X1 X3 X

2

( | )E Y XX

: slope coefficient: Change in E(Y|X)

for a change in x.

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|Xi) = conditional mean i.e. does not describe every observation– Yi = E(Y|Xi) + ui

– ui represents the deviation of each individual observation from the conditional mean

• Yi = E(Y|Xi) + ui

• Yi = 1+2 Xi + ui

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

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 ui2 i.e. min (u1

2 + u22+u3

2+…+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

• 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.

S(1 , 2) = Ni (Yi - 1-2X )

2

1 21 2

11

1 21 2

12

( , )2

( , )2 ( )

N

i iiN

i i ii

X

X

SY

SX Y

To find the minimum set these equal to zero. b1 , b2 are the solutions to these equations when they are set = 0:

1 21

1 21

2 0

2 ( ) 0

N

i iiN

i i ii

XY b b

X Y b b X

12

2

1

)(

)

( )

(

N

i ii

N

ii

x

x

x y yb

x

1 2b y b x

An Explanation

• b1, b2 are the Ordinary Least Squares (OLS) 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;

OLS in stataregress cons nmwage

Source | SS df MS Number of obs = 1330-------------+------------------------------ F( 1, 1328) = 605.97 Model | 98124170.1 1 98124170.1 Prob > F = 0.0000 Residual | 215041332 1328 161928.714 R-squared = 0.3133-------------+------------------------------ Adj R-squared = 0.3128 Total | 313165502 1329 235639.956 Root MSE = 402.4

------------------------------------------------------------------------------ cons | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- nmwage | .7562304 .0307205 24.62 0.000 .6959644 .8164964 _cons | 62.47876 25.9165 2.41 0.016 11.63701 113.3205------------------------------------------------------------------------------

Estimated coef

i ui2

The Answer

• The regression gives us a measure of the MPC• The OLS estimate of the MPC is 0.756• 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= 62.47876 + 0.7562304*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 Ui

Predicted Consumption

Y E(Y|X)=1+2X

Y4 u4

Y1 u1

Y3 u3

Y2 u2

1 X2 X1 X3 X4 X

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 (Ui)

• 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

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Distribution of Y for two different “slices” of X

f(Y|X)

f(Y|X=900)

y|x=900 consumption Probability distribution of expenditure given income = 900 and income = 1200

f(Y|X=900) f(Y|X=1200)

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<1100 &nmwage >900, norm• All OLS does is draw a line through all the means• Imagine laying all these distributions side by side

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normal cons

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Graphs by rich

The “Slice” Around nmwage=10000

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Distribution of Yf(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– Yi = 1+2 Xi + ui

– Where E(Y|Xi)= 1+2 Xi

– And ui ~N(0,s2)