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Econ 5420: Cross Section: Introduction

Prof. Jason Blevins

Department of Economics

The Ohio State University

What is Econometrics?

Model ⇐⇒ Econometrics ⇐⇒ Data

Figure 1: What is econometrics?

• Literally “economic measurement”

• Quantitative analysis of economic problems

• Application of statistical methods to connect theoretical

economic models to data

• Bridges abstract economic theory and real-world human

economic activity

• Notion of a “true model”

Relevance

• Academic research • Every field of economics

• With the exception of pure theory

• Government • Procurement auctions (e.g., FCC spectrum auctions)

• Assignment of scarce resources (e.g., timber and spectrum

auctions)

• Antitrust

• Environmental regulation

• Business • Estimate demand for a new product

• Forecasting sales

• Pricing financial assets

• Search engine advertisements

Non-ideal data

• Traditional statistics: controlled experiments

• Economists rarely have this luxury

• Much of econometrics focuses on statistical analysis of

data under the non-ideal circumstances inherent in

measuring economic interactions.

Quote: Ragnar Frisch

[T]here are several aspects of the quantitative

approach to economics, and no single one of these

aspects, taken by itself, should be confounded with

econometrics. Thus, econometrics is by no means

the same as economic statistics. Nor is it identical

with what we call general economic theory, although

a considerable portion of this theory has a definitely

quantitative character. Nor should econometrics be

taken as synonomous with the application of

mathematics to economics. . . . It is the unification of

all three that is powerful. And it is this unification that

constitutes econometrics.

–Frisch (1933)

Roles of Econometrics

In light of these definitions, it is clear that econometrics is used

for:

1. quantifying economic relationships (estimation),

2. testing economic theories, and

3. prediction and forecasting.

Demand Example

Example

Let Q denote the quantity demanded of a particular good. We

expect that Q should depend on P, the price of the good itself,

Ps, the price of a substitute good, and Yd, disposable income.

Without saying more about the specific relationships between

these variables, we can represent this notion using the abstract

functional relationship

Q = f(P, Ps, Yd).

Typically, we would expect demand (Q) to decrease with P

and increase with Ps.

Examples of the Roles of Econometrics

Examples of the three roles applied to the demand model

Q = f(P, Ps, Yd):

1. Quantifying economic relationships: In a linear model

Q = β1 + β2P + β3Ps + β4Yd,

what are the values of the β1, β2, β3, and β4?

2. Testing economic theories: Is the good a normal good?

3. Prediction and forecasting: How many units would be

demanded, given hypothetical values of prices and

income?

Estimation: Quantifying Uncertainty

Q

Yd

(a) Small sample

Q

Yd

(b) Large sample

Q

Yd

(c) Low variance

Figure 2: Statistical significance.

A Note on Causality

• Correlation does not imply causation!

• Suppose we observe that when large numbers of people

carry umbrellas to work it tends to rain.

• Obviously, carrying umbrellas do not cause it to rain!

• This is of course just a correlation, and the causation runs

in the opposite direction.

• For the purposes of prediction, perhaps the number of

umbrellas is a good predictor of rain, but for interpretation

of the underlying process it is nonsensical.

Another Note on Causality

Example

A study of traffic accidents due to alcohol examined police

reports of traffic accidents. For each, researchers recorded

whether the driver had consumed alcohol and whether or not

the report noted an empty beer container in the vehicle.

Statement A: A recent study has found that drinking beer while

driving may lead to increased risk of an accident.

Statement B: A recent study has found that empty beer

containers in cars may lead to increased risk of an accident.

Econometric Data

There are three basic types of econometric data:

1. Cross-sectional data: observations on different individual

units (e.g., people or firms) with no natural ordering.

2. Time series data: observations at different points in time

(e.g., monthly or yearly) with a natural ordering (time).

3. Panel data: a mixture of time series and cross sectional

data consisting of observations on multiple individuals

(unordered) at different points in time (ordered).

Cross-Sectional Data

Example

A cross-sectional dataset on different students including age,

GPA, and hours studied per week:

Student Age GPA Hours

1 20 3.4 10

2 21 3.1 5

3 19 3.9 12 ...

... ...

...

Time Series Data: Example

Example

A time series dataset consisting of semester-by-semester

observations on a single student:

Semester Age GPA Hours

Fall 2013 18 3.3 7

Spring 2014 18 3.8 5

Fall 2014 19 3.5 12 ...

... ...

...

Panel Data: Example

Example

A panel dataset with observations on multiple students across

multiple semesters:

Student Semester Age GPA Hours

1 Fall 2013 18 3.3 7

1 Spring 2014 18 3.8 5

1 Fall 2014 19 3.5 12 ...

... ...

... ...

2 Fall 2013 20 3.1 10

2 Spring 2014 20 2.9 10

2 Fall 2014 21 3.4 18 ...

... ...

... ...

Mean, Variance, and Standard Deviation

• Three important features of random variables: mean,

variance, and standard deviation.

• Referred to as “moments” of a distribution.

• Forumla depends on whether the distribution is discrete

(sum) or continuous (integral).

Expected Value

The expected value or mean of a discrete random variable is

the sum of all possible outcomes weighted by the probability

that each outcome occurs.

• Discrete random variable Z takes on a countable number

of outcomes.

• Let k denote the number of outcomes and z1, z2, . . . , zk are denote the values of the outcomes.

• Let P(z1), P(z2), . . . , P(zk) denote the probabilities

associated with each outcome.

• The expected value of Z, denoted E[Z] or µ is

µ = E[Z] = k∑ i=1

P(zi)zi = P(z1)z1 + · · · + P(zk)zk

Expected Value

Example

An individual makes a bet with the possibility of losing $1.00,

breaking even, winning $3.00, or winning $5.00. Let Z be a

random variable representing the winnings.

Outcome -$1.00 $0.00 $3.00 $5.00

Probability 0.30 0.40 0.20 0.10

The mean outcome is:

µ = E[Z] = (0.3 × −1) + (0.4 × 0) + (0.2 × 3) + (0.1 × 5) = −0.3 + 0.6 + 0.5 = 0.8

Therefore, the expected payoff for this bet is $0.80.

Variance

The variance of a discrete random variable Z, denoted Var(Z)

or σ2, is a measure of the variability of the distribution and is

defined as

σ2 = Var[Z] = E[(Z − µ)2] = k∑ i=1

P(zi)(zi − µ)2.

The variance is the expected value of (Z − µ)2 which is the anticipated average value of the squared deviations of Z from

the mean.

Variance: Example

Example

Returning to the example, even though the odds are favorable

one might also be concerned about the possibility of extreme

outcomes. The variance of the winnings is

σ2 = [0.3 × (−1 − 0.8)2] + [0.4 × (0 − 0.8)2] + [0.2 × (3 − 0.8)2] + [0.1 × (5 − 0.8)2]

= [0.3 × (−1.8)2] + [0.4 × (−0.8)2] + [0.2 × (2.2)2] + [0.1 × (4.2)2]

= [0.3 × 3.24] + [0.4 × 0.64] + [0.2 × 4.84] + [0.1 × 17.64] = 0.972 + 0.256 + 0.968 + 1.764

= 3.96.

Standard Deviation

The standard deviation, denoted σ, is the square root of the

variance.

Example

In our example, the variance was σ2 = 3.96 so the standard

deviation is

σ = √ 3.96 ≈ 1.99

Sample Statistics

In contrast to the population mean, the sample average or

sample mean is a sum over n sampled values (called

realizations) of a random variable where all of the weights are

all equal to 1/n. Let {Z1, Z2, . . . , Zn} denote a sample of size n from the distribution of Z. The sample average of

{Z1, Z2, . . . , Zn} is

Z̄ = 1

n

n∑ i=1

Zi = 1

n Z1 + · · · +

1

n Zn.

The sample variance is defined similarly:

s2 = 1

n

n∑ i=1

(Zi − Z̄)2.

Normal Distribution

• Mean, variance, and standard deviation are can be

illustrated intuitively with the normal distribution.

• Write N(µ, σ2) to denote the normal distribution with

mean µ and variance σ2.

• The standard deviation is defined as the square root of the

variance, σ in this case.

Normal Distribtion

0.00

0.04

0.09

0.14

0.18

0.22

0.27

0.31

0.36

0.40

0.45

-9.0 -7.2 -5.4 -3.6 -1.8 0.0 1.8 3.6 5.4 7.2 9.0

P ro

b a b ili

ty d

e n si

ty

x

Figure 3: N(0, 1)

Normal Distribtion

0.00

0.04

0.09

0.14

0.18

0.22

0.27

0.31

0.36

0.40

0.45

-9.0 -7.2 -5.4 -3.6 -1.8 0.0 1.8 3.6 5.4 7.2 9.0

P ro

b a b ili

ty d

e n si

ty

x

Figure 4: N(1, 2)

Simple Regression

• Ordinary Least Squares (OLS) with a single independent

variable.

• The theoretical model of interest is the linear model

Yi = β0 + β1Xi + ui.

• From this equation, we seek to use the information

contained in a dataset of observations (Xi, Yi) to estimate

the values of β0 and β1, which we call β̂1 and β̂2.

• The fitted values are

Ŷi = β̂0 + β̂1Xi

• The residuals are the differences between the fitted values

Ŷi and the observed values Yi:

ûi ≡ Yi − Ŷi.

The Geometry of Simple Regression

0

Y

XX1 X2

Y2

Ŷ2

Ŷ1

Y1

û2

û1

Figure 5: Estimated regression line and residuals

Simple Regression

• We need to formally define our loss function, the criteria

by which we determine whether the fit is good or not.

• OLS is founded on minimizing the sum of squared

residuals (SSR), defined as

SSR = n∑ i=1

û2i = û 2 1 + û

2 2 + · · · + û

2 n,

where n is the sample size.

• OLS estimates of β0 and β1 are defined to be the values of

β̂0 and β̂1 which, when plugged into the estimated

regression equation, minimize the SSR.

• Note that different samples (of the same size) yield

different estimates.

Why Ordinary Least Squares?

OLS is used so often for several reasons.

1. It is very straightforward to implement, both by hand and

computationally, and it is simple to work with theoretically.

2. The criteria of minimizing the squared residuals is intuitive.

3. Other nice properties: the regression line passes through

the means of X and Y, (X̄, Ȳ), the sum of the residuals is

zero, etc.

Simple Regression: Interpretation

In the regression line

Yi = β̂0 + β̂1Xi,

the coefficient on Xi represents the amount by which we

predict Yi will increase when Xi increases by one unit.

Simple Regression: Interpretation

Example

Let Yi be an individual i’s annual demand for housing in dollars

and let Xi be individual i’s annual income, also measured in

dollars.

Then β̂1 is the number of additional dollars individual i is

predicted to spend on housing when income increases by one

dollar.

The intercept, β̂0, is an individual’s predicted expenditure on

housing when income is zero.

Anscombe's Quartet: A Cautionary Tale

4 6 8 10 12 14 16 18

4

6

8

10

12

x1

y 1

4 6 8 10 12 14 16 18

4

6

8

10

12

x2

y 2

4 6 8 10 12 14 16 18

4

6

8

10

12

x3

y 3

4 6 8 10 12 14 16 18

4

6

8

10

12

x4

y 4

OLS Review: Sum of the Residuals is Zero

To see that the residuals sum to zero, we can look at the

average of the residuals. If the average is zero, so is the sum.

1

n

n∑ i=1

ûi = 1

n

n∑ i=1

( Yi − β̂0 − β̂1Xi

) =

1

n

n∑ i=1

Yi − β̂0 − β̂1 1

n

n∑ i=1

Xi

= Ȳ − β̂0 − β̂1X̄.

But recall that β̂0 = Ȳ − β̂1X̄, and so

1

n

n∑ i=1

ûi = Ȳ − β̂0 − β̂1X̄.

= Ȳ − (Ȳ − β̂1X̄) − β̂1X̄ = 0.

OLS Review: Fitted Value at Mean

Show that the regression line passes through the point (X̄, Ȳ).

We can evaluate the regression line at the point X̄ and show

that the fitted value is indeed Ȳ.

Substituting for β̂0 gives:

Ŷ = β̂0 + β̂1X

= (Ȳ − β̂1X̄) + β̂1X.

Then evaluating at the point X = X̄ gives:

Ŷ = (Ȳ − β̂1X̄) + β̂1X̄ = Ȳ.

OLS Review: Average of Predicted Values

Another property is that the average of the predicted values Ŷi equals the average of the observations Yi:

1

n

n∑ i=1

Ŷi = 1

n

n∑ i=1

(β̂0 − β̂1Xi)

= β̂0 + β̂1X̄

= (Ȳ − β̂1X̄) + β̂1X̄ = Ȳ.

References

Frisch, R. (1933). Editor’s note. Econometrica 1, 1–4.