The linear regression model: Theory and Application
1.
Statistics Lab
Rodolfo Metulini
IMT Institute for Advanced Studies, Lucca, Italy
Lesson 4 - The linear Regression Model: Theory and
Application - 21.01.2014
2.
Introduction
In the past praticals we analyzed one variable.
For certain reasons, it is even usefull to analyze two or more
variables together.
The question we want to asnwer regards what are the relations,
the causal eﬀects determining changes in a variable. Analyze if a
certain phenomenon is endogenous or exogenous.
In symbols, the idea can be represent as follow:
y = f (x1 , x2 , ...)
Y is the response, which is a function (it depends on) one or more
variables.
3.
Objectives
All in all, the regression model is the instrument used to:
measure the entity of the relations between two or more
variables: Y / X ,
and to measure the causal direction ( X −→
viceversa? )
Y or
forecast the value of the variable Y in response to some
changes in the others X1 , X2 , ... (called explanatories),
or for some cases that are not considered in the sample.
4.
Simple linear regression model
The regression model is stochastic, not deterministic.
Giving two sets of values (two variables) from a random sample of
length n: x = {x1 , x2 , ..., xi , ..xn }; y = {y1 , y2 , ..., yi , ..yn }:
Deterministic formula:
yi = β0 + β1 xi , ∀i = 1, .., n
Stochastic formula:
yi = β0 + β1 xi +
where
i
i
∀i = 1, .., n
is the stochastic component.
β1 deﬁne the slope in the relations between X and Y (See graph in
chart 1)
5.
Simple linear regression model - 2
ˆ
ˆ ˆ
We need to ﬁnd β = {β0 , β1 } as estimators of β0 and β1 .
After β is estimated, we can draw the estimated regression line,
which corresponds to the estimated regression model, as
follow:
ˆ
ˆ
yi = β0 + β1 xi
ˆ
Here, ˆi = yi − yi .
ˆ
Where yi is the i-element of the estimated Y vector, and yi is the
ˆ
i-elements of the real Y vector. (see graph in chart 2)
6.
Steps in the Analysis
1. Study the relations (scatterplot, correlations) between two or
more variables.
ˆ
ˆ ˆ
2. Estimation of the parameters of the model β = {β0 , β1 }.
ˆ
3. Hypotesis tests on the estimated β1 to verify the casual
eﬀects between X and Y
4. Robustness check of the model.
5. Use the model to analyze the causal eﬀect and/or to do
forecasting.
7.
Why linear?
It is simple to estimate, to analyze and to interpret
it likely ﬁts with most of empirical cases, in which the
relations between two phenomenon is linear.
There are a lot of implemented methods to transorm variables
in order to obtain a linear relationship (log transformation,
normalization, etc.. )
8.
Model Hypotesis
In order the estimation and the utilization of the model to be
correct, certain hypotesis must hold:
E ( i ) = 0, ∀i −→ E (yi ) = β0 + β1 xi
Omoschedasticity: V ( i ) = σi2 = σ 2 , ∀i
Null covariance: Cov ( i , j ) = 0, ∀i = j
Null covariance among residuals and explanatories:
Cov (xi , i ) = 0, ∀i, since X is deterministic (known)
Normal assumption:
i
∼ N(0, σ 2 )
9.
Model Hypotesis - 2
From the hypotesis above, follow that:
V (yi ) = σ 2 , ∀i. Y is stochastic only for the
component.
Cov (yi , yj ) = 0, ∀i = j. Since the residuals are uncorrelated.
yi ∼ N[(β0 + β1 x1 ), σ 2 ] Since also the residuals are normal in
shape.
10.
Ordinary Least Squares (OLS) Estimation
The OLS is the estimation method used to estimate the vector β.
The idea is to minimize the value of the residuals.
Since ei = yi − yi we are interested in minimize the component
ˆ
ˆ
ˆ
yi − β0 − β1 xi .
N.B.
i
ˆ
ˆ
= β0 − β1 xi , while ei = β0 − β1 xi
The method consist in minimize the sum of the square
diﬀerences:
n
i (yi
− yi )2 =
ˆ
n 2
i ei
= Min,
which is equal to solve this 2 equation system derived using
derivates.
11.
Ordinary Least Squares (OLS) Estimation - 2
n
ei2 = 0
(1)
ei2 = 0
δ/δβ0
(2)
i
n
δ/δβ1
i
After some arithmetics, we end up with this estimators for the
vector β:
β0 = y − β1 x
¯ ˆ ¯
n
¯
¯
i (yi − y )(xi − x )
β1 =
n
2
¯
i (xi − x )
(3)
(4)
12.
OLS estimators
ˆ
ˆ
OLS β0 and β1 are stochastic estimators (they have a
distribution in a sample space of all the possible estimtors
deﬁne with diﬀerent samples)
ˆ
β1 : measure the estimated variation in Y determined by a
unitary variation in X (δY /δX )
ˆ
The OLS estimators are correct (E (β1 ) = β1 ),
and they are BLUE (corrects and with the lowest variance)
13.
Linear dependency index (R 2 )
The R 2 index is the most used index to measure the linear ﬁtting
of the model.
R 2 is conﬁned in the boundary [−1, 1], where, values near to 1 (or
-1) means the explanatories are usefull to describe the changes in
Y.
Let deﬁne
SQT = SQR + SQE , or
n
i (yi
− y )2 =
¯
n
y
i (ˆi
The R 2 is deﬁned as
R2 =
n
y y 2
i (ˆi −¯)
n
y 2
i (yi −¯)
− y )2 +
¯
SQR
SQT
or 1 −
n
i (yi
− y i )2
ˆ
SQE
SQT .
Or, equivalent:
14.
Hypotesis testing on β1
The estimated slope parameter β1 is stochastic. It distributes as a
gaussian:
ˆ
β1 ∼ N[β1 , σ 2 /SSx]
We can make use of the hypotesis tests approach to investigate on
the causal relation between Y and X :
H0 : β1 = 0
H1 : β1 = 0,
where, alternative hypotesis mean causal relation.
The test is:
z=
ˆ
β1 −β1
sqrt(σ 2 /SSx)
∼ N(0, 1).
When SSx is unknown, we estimate it as : SSx =
and we use t − test with n − 1 degrees of freedom
n
i (xi
− y )2 ,
¯
15.
Forecasting within the regresion model
The question we want to answer is the following: Which is the
expected value of Y (say yn+1 ), for a certain observation that is
not in the sample?.
Suppose we have, for that observation, the value for the variable X
(say xn+1 )
We make use of the estimated β to determine:
ˆ
ˆ
yn+1 = β0 + β1 xn+1
ˆ
16.
Model Checking
Several methods are used to test the robustness of the model,
most of them based on the stochastic part of the the model: the
estimated residuals.
Graphical checks: Plot residuals versus ﬁtted values
qq-plot for the normality
Shapiro wilk test for normality
Durbin-Watson test for serial correlation
Breusch-Pagan test for heteroschedasticity
Moreover, the leverage is used to evaluate th importance of each
observation in determining the estimated coeﬃcients β.
The Stepwise procedure is used to choice between diﬀerent model
speciﬁcations.
17.
Model Checking using estimated residuals - Linearity
An example of departure from the linearity assumption: we can
draw a curve (not a horizontal line) to interpolate the points
Figure: residuals (Y) versus estimated (X) values
18.
Model Checking using estimated residuals Omoscedasticity
An example of departure from the omoschedasticity assumption
(the estimated residuals increases as the predicted values
increase)
19.
Model Checking using estimated residuals - Normality
An example of departure from the normality assumption: the
qq-points do not follow the qq-line
Figure: residuals (Y) versus estimated (X) values
20.
Model Checking using estimated residuals - Serial
correlation
An example of departure from the serial incorrelation assumption:
the residual at i depend on the value at i − 1
21.
Homeworks
1. Using cement data (n = 13), determine the β0 and β1
coeﬃcients manually, using OLS formula at page 11, of the
model y = β0 + β1 x1
2. Using cement data, estimate the R 2 index of the model
y = β0 + β1 x1 , using formula at page 13.
22.
Charts - 1
Figure: Slope coeﬃcient in the linear model
23.
Charts - 2
Figure: Fitted (line) versus real (points) values
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