Econometrics is the quantitative analysis of economic phenomena using statistical methods. It has three main uses: 1) describing economic relationships, 2) testing economic theories, and 3) forecasting future economic trends. Regression analysis is a statistical technique used in econometrics to model relationships between variables. It involves estimating coefficients for independent variables that quantify their impact on a dependent variable. The estimated regression equation relates the dependent variable to the independent variables plus an error term. This allows for variations not explained by the independent variables.

1. A. H. Studenmund
A Practical Guide To
Using Econometrics
Chapter 1

2. β 1-2
What Is Econometrics?
• Can mean different things to different people.
• Formally defined:
Econometrics is the quantitative
measurement and analysis of actual
business and economic phenomena.
• Econometrics attempts to quantitatively bridge gap
between economic theory and the real world.
• In practice, econometrics has three major uses.

3. β 1-3
What Is Econometrics? (continued)
Use 1: Describing economic reality
• Econometrics can quantify and measure marginal effects
and estimate numbers for theoretical equations.
• For example, consumer demand for a product often can
be thought of as a relationship between the quantity
demanded (Q) and its price (P), the price of a substitute
(Ps), and disposable income (Yd).

4. β 1-4
What Is Econometrics? (continued)
• Written as a theoretical equation:
• Theory suggests β3 should be positive for most goods.
• Econometrics allows us to estimate that relationship
using past consumption, income and prices:
• If Yd increases by one unit, Q increases by 0.23.
• 0.23 is called an estimated regression coefficient.
Q = b0 +b1 P+b2 PS +b3 Yd (1.1)
Q = 27.7+0.11P+0.03PS +0.23Yd (1.2)

5. β 1-5
What Is Econometrics? (continued)
Use 2: Testing hypotheses about economic theory
and policy.
• Much of economics involves building theoretical models
and testing them against evidence.
• Hypothesis testing is a vital part of that process.
• You could test the hypothesis that the product in
Equation (1.1) is a normal good.
β
Q = b0 +b1 P+b2 PS +b3 Yd (1.1)

6. β 1-6
What Is Econometrics? (continued)
• Since β3 was estimated to be 0.23, the evidence seems
to support the hypothesis.
• But the “statistical significance” of the estimate would
have to be investigated before such a conclusion could
be justified.
Q = 27.7+0.11P+0.03PS +0.23Yd (1.2)

7. β 1-7
What Is Econometrics? (continued)
Use 3: Forecasting future economic activity
• The most difficult use of econometrics is to forecast or
predict the future using past data.
• Economists use econometrics to forecast a variety of
variables (GDP, sales, inflation, etc.).
• Accuracy of forecasts depends in large measure on the
degree to which the past is a good guide to the future.
• To the extent econometrics can shed light on the future,
leaders will be better equipped to make decisions.

8. β 1-8
What Is Econometrics? (continued)
• There are different approaches to quantitative work.
• Different academic disciplines use different techniques
because they face different problems.
• Sir Clive Granger, Nobel Laureate, noted:
“We need a special field called econometrics,
and textbooks about it, because it is generally
accepted that economic data possess certain
properties that are not considered in standard
statistics texts or are not sufficiently emphasized
there for use by economists.”

9. β 1-9
What Is Econometrics? (continued)
• Different approaches also make sense in economics.
• Forecasting uses different techniques than econometric
modeling for descriptive purposes.
• To put into context, consider the steps to
nonexperimental quantitative research:
1. Specifying the models or relationships to be studied.
2. Collecting the data needed to quantify the models.
3. Quantifying the models with the data.
• The approach chosen is left to individual but should be
justifiable.

10. β 1-10
What Is Regression Analysis?
• It is a statistical technique that attempts to “explain”
movements in one variable, the dependent variable, as
a function of movements in a set of other variables,
called the independent (or explanatory) variables,
through the quantification of one or more equations.
• For example, in equation (1.1):
dependent variable: Q
independent variables: P, Ps, and Yd
Q = b0 +b1 P+b2 PS +b3 Yd (1.1)

11. β 1-11
What Is Regression Analysis? (continued)
• Economists are interested in cause-and-effect.
• Don’t be deceived by the words “dependent” and
“independent” variables.
• Regression results cannot prove causality!
• For example, if variables A and B are related statistically,
then:
-A might “cause” B.
-B might “cause” A.
-Some third factor might “cause” both.
-The relationship might have happened by chance.

12. β 1-12
What Is Regression Analysis? (continued)
• The simplest single-equation linear model is:
• Y is the dependent variable
• X is the independent variable
• β’s are coefficients
β0 is the constant or intercept term
β1 is the slope coefficient
• The slope coefficient, β1, indicates the amount Y will
change if X increases by one unit.
Y = b0 +b1 X (1.3)

13. β 1-13
What Is Regression Analysis? (continued)
• The slope coefficient, β1, indicates the amount Y will
change if X increases by one unit.
• If linear regression techniques are going to be applied to
an equation, that equation must be linear.
• An equation is linear if plotting the function in terms of X
and Y generates a straight line.
(Y2 -Y1)
(X2 - X1)
=
DY
DX
= b1

14. β
What Is Regression Analysis? (continued)

15. β 1-15
What Is Regression Analysis? (continued)
• Even if much of the variation in Y is caused by X, there
is almost always variation that comes from other
sources.
• A stochastic error term is added to a regression
equation to account for variation in Y that cannot be
explained by the included X(s).
• This is usually notated by adding an epsilon (ε) to the
regression equation:
Y = b0 +b1 X +e (1.4)

16. β 1-16
What Is Regression Analysis? (continued)
• Equation (1.4) can be thought of as having two parts:
1. Deterministic: β0 + β1X
2. Stochastic (or random): ε
• The deterministic component indicates the value of Y
that is determined by a given value of X.
• The deterministic can be thought of as the expected
value of Y given X:
E(Y | X) = b0 +b1 X (1.5)

17. β 1-17
What Is Regression Analysis? (continued)
• The stochastic term (the error term, ε) “catches” the
sources of variation that the deterministic part does not.
• There are at least four sources of variation in Y not
captured by the included X(s):
1. Influences omitted from the equation
2. Measurement error in the dependent variable
3. The true theoretical equation has a different
functional form than the one chosen for the
regression
4. Human behavior can be unpredictable and
purely random

18. β 1-18
What Is Regression Analysis? (continued)
Example: Aggregate consumption function
• Possible sources of error?
1. Consumer uncertainty hard to measure (omitted
variable)
2. Observed consumption different than actual
consumption (sampling error)
3. The consumption function might not be linear
(different functional form)
4. Some random event (purely random)
Consumption = b0 +b1 DisposableIncome+e

19. β 1-19
What Is Regression Analysis? (continued)

20. β 1-20
What Is Regression Analysis? (continued)
• Notation needs to be extended to allow for more than
one independent variable and reference specific
observations.
• First, extend notation to reference specific observations:
where:
Yi = the ith observation of the dependent variable
Xi = the ith observation of the independent variable
εi = the ith observation of the stochastic error term
β0, β1 are the regression coefficients
N is the number of observations
Yi = b0 +b1 Xi +ei (1.7)
(i =1,2,..., N)

21. β 1-21
What Is Regression Analysis? (continued)
• Second, extend notation to allow for more than one
independent variable.
• If we define:
X1i = the ith observation of the first independent
variable
X2i = the ith observation of the second independent
variable
X3i = the ith observation of the third independent
variable
• Then, all three variables can be expressed as
determinants of Y.

22. β 1-22
What Is Regression Analysis? (continued)
• These extensions result in a multivariate linear
regression model:
• Each slope coefficient gives the impact on Y of a 1 unit
increase in its X, holding constant other included X’s.
• For example, if X2 increases by 1 unit, Y increases by β2
holding X1 and X3 constant.
• If a variable is not included in an equation, its impact on
Y is not held constant.
Yi = b0 +b1 X1i +b2 X2i +b3 X3i +ei (1.8)

23. β 1-23
What Is Regression Analysis? (continued)
Example: Weight as a function of height
• Each value of i represents an individual in the sample.
• If you select four individuals (Woody, Lesley, Bruce, and
Mary), then you could write out an equation for each:
Weighti = b0 +b1 Heighti +ei (1.9)
Weightwoody = b0 +b1 Heightwoody +ewoody
Weightlesley = b0 +b1 Heightlesley +elesley
Weightbruce = b0 +b1 Heightbruce +ebruce
Weightmary = b0 +b1 Heightmary +emary

24. β 1-24
What Is Regression Analysis? (continued)
• Each individual has their own height and weight.
• Random events impact people differently.
• To account for these random differences each individual
needs their own value of the error term (εi).
• Note that the regression coefficients (the β’s) don’t
vary by individual.
• Rather, the β’s apply to the whole sample.

25. β 1-25
What Is Regression Analysis? (continued)
Example: What influences wages?
• Wage (WAGE) of worker is dependent variable
• Possible independent variables?
experience (EXP), education (EDU), gender (GEND)
• Redefine variables in Equation (1.8):
Y = WAGE X2 = EDU
X1 = EXP X3 = GEND
• Substituting these into Equation (1.8):
WAGEi = b0 +b1 EXPi +b2 EDUi +b3 GENDi +ei (1.10)

26. β 1-26
What Is Regression Analysis? (continued)
• What is the meaning of β1 in equation (1.10)?
• It is the impact on wages of an additional year of
experience holding constant education and gender.
• General multivariate linear regression model with K
variables:
Yi = b0 +b1 X1i +b2 X2i +...+bK XKi +ei (1.11)

27. β 1-27
The Estimated Regression Equation
• The quantified version of a theoretical regression
equation is the estimated regression equation.
Theoretical:
Estimated:
• (read “Y-hat”) is the estimated or fitted value of Yi
• Put another way:
• The closer the are to Yi’s, the better the “fit.”
ˆ
Yi =103.40+6.38Xi
Yi = b0 +b1Xi +ei
ˆ
Yi
E[Yi | Xi ]= ˆ
Yi
ˆ
Yi 's
(1.12)
(1.13)

28. β 1-28
The Estimated Regression Equation (continued)
The difference between and is the residual (ei).
• Mathematically:
• Note the difference between ei and εi:
• The residual (ei) can be thought of as an estimate of the
error term (εi ).
• Figure 1.3 graphically displays these concepts.
ˆ
Yi
ei =Yi - ˆ
Yi (1.15)
ei =Yi -E(Yi | Xi ) (1.16)
Yi

29. β 1-29
The Estimated Regression Equation (continued)

30. β 1-30
The Estimated Regression Equation (continued)
True Regression
Equation
Estimated Regression
Equation
b̂0
b̂1
ei
b0
b1
ei
• The estimated regression model can be extended by
adding additional X’s.
ˆ
Yi = b̂0 +b̂1 X1i +b̂2 X2i +...+b̂K XKi
(1.17)

31. β 1-31
A Simple Example of Regression Analysis
• You’ve accepted a job as a weight guesser at Magic Hill.
• You hypothesize the following theoretical relationship.
where:
Yi = the weight (in pounds) of ith customer
Xi = the height (in inches above 5 ft) of ith customer
εi = the value of the stochastic error term for the
ith customer
• The estimated equation:
Yi = b0 +b
+
1 Xi +ei
(1.18)
EstimatedWeight =103.40+6.38Height(> 5ft) (1.19)

32. β 1-32
A Simple Example (continued)

33. β 1-33
A Simple Example (continued)

34. β 1-34
Using Regression to Explain Housing Prices
• Want to measure the impact of house size on price.
• Theoretical model:
where:
PRICEi = the price (in thousands of $) of the ith house
SIZEi = the size (in square feet) of the ith house
εi = the value of the stochastic error term for the
ith house
• The estimated equation:
PRICEi = b0 +b
+
1 SIZEi +ei
PRˆ
ICEi = 40.0+0.138SIZEi
(1.20)
(1.21)

35. β 1-35
Explain Housing Prices (continued)

36. β 1-36
Explain Housing Prices (continued)
• What does = 40.0 mean?
• It is the estimate of the constant or intercept term (β0).
• Take care in interpreting (more about that in Chapter 7).
• What does = 0.138 mean?
• It is the estimate of the coefficient of SIZE (β1).
• Interpretation: if the size of a house increases by 1
square foot, the estimated price of the house will
increase $138.
b̂0
b̂1

37. β 1-37
Explain Housing Prices (continued)
• What does the model predict for a 1600 sqft house?
• Since PRICE is in thousands, estimated price is $260,800
• Perhaps the price of a house is influenced by more than
just the size of the house? (This example is revisited in
Chapter 11).
ˆ 40.0 0.138(1600) 260.8
i
PRICE = + =

38. β
CHAPTER 1: the end
β