Intro to econometrics

Introduction to Econometrics
Gaetan “Guy” Lion
June 2015
1

Table of Content
1) Linear Regression.
2) Multiple Regression. Multiple Regression as an
Optimization.
3) Building an Econometrics model. Stepwise
Regression, Autoregressive Regression.
4) Model testing. Multicollinearity, Autocorrelation,
Heteroskedasticity, Robust Standard Errors, Outliers testing,
Normality, Scenario testing.
2

The Basic Linear Regression Equation
4
Y = Constant + bX + Error term
Y is the dependent
variable we want
to estimate or
model.
b is a coefficient that
multiplies the
independent variable
X. It is called the
Slope of X. It reflects
Xs influence on Y, the
dependent variable.
X is the
independent
variable that helps
us in estimating Y.
Constant also
called the
Intercept is the
value of Y when X
is equal to zero.
Error term also
called Residual is the
difference between
the actual value of Y
and the estimated
value of Y derived
from: Const. + bX
A Regression Model allows us to
estimate and explain the
behavior of a variable Y using an
independent variable X.

Let’s estimate Economic Growth using
Home Price changes
5
Y X
R. GDP
Home
price chg.
2000 4.1% 4.1%
2001 1.0% 5.8%
2002 1.8% 7.6%
2003 2.8% 7.3%
2004 3.8% 8.1%
2005 3.3% 12.8%
2006 2.7% 2.1%
2007 1.8% -2.9%
2008 -0.3% -9.2%
2009 -2.8% -11.9%
2010 2.5% 0.1%
2011 1.6% -4.5%
2012 2.3% 6.5%
2013 2.2% 11.4%
2014 2.4% 5.8%
Within our data set Real GDP growth
(annual) is the dependent variable Y. And,
annual Home Price change is the
independent variable.
Linear Regression allows us to explore
how well we can estimate Real GDP
growth, if we know the Home Price chg.

Excel Scatter Plot = Regression the easy way
6
H:AbilitiesProjects2015EconometricsBasics.xlsxLinear regression
Running a Linear Regression visually is very easy in three easy steps:
1) Do a Scatter Plot with your independent variable X (Home price chg.) on the X-axis and
your dependent variable Y (Real GDP growth) on the Y-axis;
2) Add a Trendline to your Scatter Plot. That is actually your Regression line that best fit the
data;
3) Format your Trendline by adding the actual regression equation and the R^2 measure
that tells how much the variable X explains of the variance of variable Y.
The regressed equation solution: Real GDP Growth = 1.44% + 0.177(Home price chg.)
Step 1: Do a Scatter Plot with X var. on X-axis; Y var. on Y-Axis Step 2: Add a Trendline Step 3: Format Trendline by adding equation and R^2
-4%
-3%
-2%
-1%
0%
1%
2%
3%
4%
5%
-15% -10% -5% 0% 5% 10% 15%
RGDPgrowth
Home price change
R GDP growthvs Home Price change
-4%
-3%
-2%
-1%
0%
1%
2%
3%
4%
5%
-15% -10% -5% 0% 5% 10% 15%
RGDPgrowth
Home price change
y = 0.1771x+0.0144
R² = 0.5674
-4%
-3%
-2%
-1%
0%
1%
2%
3%
4%
5%
-15% -10% -5% 0% 5% 10% 15%
RGDPgrowth
Home price change

The Geometry of Linear Regression
7
Constant or Intercept = value of Y when X = 0
Beta coefficient or Slope = Chg. in Y/Chg. in X
-4%
-3%
-2%
-1%
0%
1%
2%
3%
4%
5%
-15% -10% -5% 0% 5% 10% 15%
RGDPgrowth
Home price change
Intercept
-4%
-3%
-2%
-1%
0%
1%
2%
3%
4%
5%
-15% -10% -5% 0% 5% 10% 15%
RGDPgrowth
Home price change
Chg. in Y
Chg. in X
H:AbilitiesProjects2015EconometricsBasics.xlsxLinear regression

The Arithmetic of Linear Regression
8H:AbilitiesProjects2015EconometricsBasics.xlsxLinear regression
Y X
R. GDP
Home
price chg. XY X^2 Y^2
2000 4.1% 4.1% 0.2% 0.2% 0.2%
2001 1.0% 5.8% 0.1% 0.3% 0.0%
2002 1.8% 7.6% 0.1% 0.6% 0.0%
2003 2.8% 7.3% 0.2% 0.5% 0.1%
2004 3.8% 8.1% 0.3% 0.7% 0.1%
2005 3.3% 12.8% 0.4% 1.6% 0.1%
2006 2.7% 2.1% 0.1% 0.0% 0.1%
2007 1.8% -2.9% -0.1% 0.1% 0.0%
2008 -0.3% -9.2% 0.0% 0.8% 0.0%
2009 -2.8% -11.9% 0.3% 1.4% 0.1%
2010 2.5% 0.1% 0.0% 0.0% 0.1%
2011 1.6% -4.5% -0.1% 0.2% 0.0%
2012 2.3% 6.5% 0.2% 0.4% 0.1%
2013 2.2% 11.4% 0.3% 1.3% 0.0%
2014 2.4% 5.8% 0.1% 0.3% 0.1%
Average 1.9% 2.9% 0.1% 0.6% 0.1%
Values used in Numerator 1.9% 2.9% 0.1%
Values used in Denominator 2.9% 0.6%
Calculating the Slope b :
Numerator: avg(XY) - avgX*avgY
Y = C + b X Denominator: avgX^2 - (avgX)^2
Numerator: 0.1%
Denominator: 0.5%
b 0.177
Calculating the Constant or Intercept:
C or Constant = avgY - b avgX
C 1.44%

A 2nd Arithmetic Approach
Slope b = Covariance (X, Y)/Variance(X)
9
Covar(X,Y) Var(X)
R GDP Home price A B A x B B^2
Y X Y - Avg. X - Avg.
2000 4.1% 4.1% 2.1% 1.2% 0.0% 0.0%
2001 1.0% 5.8% -1.0% 2.9% 0.0% 0.1%
2002 1.8% 7.6% -0.2% 4.7% 0.0% 0.2%
2003 2.8% 7.3% 0.9% 4.4% 0.0% 0.2%
2004 3.8% 8.1% 1.8% 5.3% 0.1% 0.3%
2005 3.3% 12.8% 1.4% 9.9% 0.1% 1.0%
2006 2.7% 2.1% 0.7% -0.8% 0.0% 0.0%
2007 1.8% -2.9% -0.2% -5.8% 0.0% 0.3%
2008 -0.3% -9.2% -2.2% -12.0% 0.3% 1.4%
2009 -2.8% -11.9% -4.7% -14.8% 0.7% 2.2%
2010 2.5% 0.1% 0.6% -2.7% 0.0% 0.1%
2011 1.6% -4.5% -0.3% -7.4% 0.0% 0.5%
2012 2.3% 6.5% 0.4% 3.6% 0.0% 0.1%
2013 2.2% 11.4% 0.3% 8.6% 0.0% 0.7%
2014 2.4% 5.8% 0.4% 2.9% 0.0% 0.1%
Average 1.9% 2.9% 1.3% 7.3% Sum
0.09% 0.49% Sum/n
Slope = Covar(X,Y)/Var(X)
Numerator 0.09%
Denominator 0.49%
Slope 0.177
H:AbilitiesProjects2015EconometricsBasics.xlsxLinear regression2

Linear Regression Excel Basics
10
R GDP Home price
Y Y est. X
2000 4.1% 2.2% 4.1%
2001 1.0% 2.5% 5.8%
2002 1.8% 2.8% 7.6%
2003 2.8% 2.7% 7.3%
2004 3.8% 2.9% 8.1%
2005 3.3% 3.7% 12.8%
2006 2.7% 1.8% 2.1%
2007 1.8% 0.9% -2.9%
2008 -0.3% -0.2% -9.2%
2009 -2.8% -0.7% -11.9%
2010 2.5% 1.5% 0.1%
2011 1.6% 0.6% -4.5%
2012 2.3% 2.6% 6.5%
2013 2.2% 3.5% 11.4%
2014 2.4% 2.5% 5.8%
Basic formulas
SLOPE() 0.177
INTERCEPT() 1.44%
RSQ () 0.567
STYX() 1.2%RSQ() = R Square. is the
square of correlation
between Y and Y est. It
tells how well the model’s
estimates fit the actual
data. It also tells what is
the % of the dependent
variable’s variance
explained by the model.
This value ranges from 0 (a
terrible model that does
not fit or explain the data)
to 1 (a perfect model that
fits the data identically and
explains 100% of the
variance of the dependent
variable).
STYX () = Standard Error
of Model. Assuming
that the Errors are
normally distributed,
one can assume that
about 2/3ds of data
observations fall within +
or – 1 Standard Error
from the model’s
estimate. And, 95% of
them fall within + or –
1.96 Standard Errors
away from the model’s
estimate.

Linear Regression with LINEST()
11H:AbilitiesProjects2015EconometricsBasics.xlsxLinear regression2
LINEST () Regression with one ind. Variable
X
Home pr. Intercept
Coefficient 0.177 1.44%
SE of Coeff 0.043 0.32%
R Square St. Error 0.567 1.2%
F Stat df Residual 17.05 13
SS Regres SS Residual 0.0023 0.0017
Rearranging LINEST() results in a standard format
Coeffic. St. Error t Stat P-value
Intercept 1.44% 0.32% 4.44 0.0007
Home price 0.177 0.043 4.13 0.0012
The LINES() formula generates a lot of info
including the Standard Error of the specific
regression coefficient(s).
This allows us to evaluate whether Home price
chg. is a good explanatory variable to keep in
model. Is it statistically significant? What is
the probability that its regression coefficient is
not different from 0?
Let’s answer those questions. By dividing
Home price’s reg. coefficient by its Standard
Error we get its t Stat: 0.177/0.043 = 4.13.
In turn, we can calculate the probability that this regression coefficient is not different
from 0 using the TDIST() function. Its arguments include: t Stat, df Residual, and # of
tails you want to test four (which is always 2 in regressions). TDIST (4.13, 13, 2) = 0.0012
which is essentially 0, meaning there is a near 0% probability that this regression
coefficient could be 0. We can be nearly 100% confident, this reg. coefficient is
different than 0. Thus, we are confident Home price chg. does belong in this model and
is a good explanatory variable to explain and estimate Real GDP growth.

Depicting 95% Confidence Interval
12
-4.0%
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0.0%
1.0%
2.0%
3.0%
4.0%
5.0%
6.0%
7.0%
Real GDPGrowthActual, Est, 95% C.I.
Y Y est. CI Low CI High
-4.0%
-3.0%
-2.0%
-1.0%
0.0%
1.0%
2.0%
3.0%
4.0%
5.0%
6.0%
7.0%
-15.0% -10.0% -5.0% 0.0% 5.0% 10.0% 15.0%
RealGDPgrowth
Home Price change
Real GDP Growth vs Home price chg, 95% C.I.
Y Y est. CI Low CI High
H:AbilitiesProjects2015EconometricsBasics.xlsxLinear regression2
Depicting 95% C.I. over time Depicting 95% C.I. vs. Home Price chg.
A 95% Confidence Interval means that we would expect that about 1 observation out of 20
would fall outside the Confidence Interval. The graphs look about right. We have only 15
observations. But, two of them are just within the C.I. All others are well within.
The C.I. Low range is 1.96 Standard Error of the model below Y estimate. The C.I. High range
is 1.96 Standard Error above Y estimate.

2) Multiple Regression as an Optimization
13
Those two methods are identical. The only difference is that the
Regression statistical output gives you a lot of very valuable
information about a model that Optimization does not.

The Basic Multiple Regression Equation
14
Y = Constant + b1X1 + b2X2 + Error term
Y is the dependent
variable we want
to estimate or
model.
b1 is a coefficient
that multiplies the
independent variable
X1. It reflects X1’s
influence on Y, the
dependent variable.
X1 is the 1st
independent
variable that helps
us in estimating Y.
X2 is the 2nd
independent
variable that helps
us in estimating Y.
Constant also
called the
Intercept is the
value of Y when X1
and X2 are equal
to zero.
b2 same
explanation
as b1.
Error term also
called Residual is the
difference between
the actual value of Y
and the estimated
value of Y derived
from: Const. + b1X1
+ b2X2.
Such Regressions can
have many more
independent variables
X3, X4, X5, …

The objective of such modeling
15
Find the Constant and b1 and b2 coefficients so as to minimize
the sum of the square of the Error terms or Residuals. That is
why Regression is called (OLS) Regression. OLS means
Ordinary Least Square (minimizing the Square of the
Residuals). That is specifically an optimization process.
Modify Constant, b1, and b2
Minimize sum of square
of Error terms.
As described Multiple Regression is actually an Optimization.

An Optimization Example
16
Real GDP Growth = Constant + b1Home Price chg + b2S&P 500 chg + Error term
We are going to model or estimate annual Real GDP Growth with
two independent variables: Home Price yearly change and S&P 500
yearly change.

Optimization starting point
17
What to change
Constant 2%
b 1 Home price 0.1
b 2 S&P 500 0.1
Y Y est. Error Error^2 X1 X2
R. GDP Estimate Residual Residual^2
Home
price S&P 500
2000 4.1% 3.2% -0.9% 0.0% 4.1% 7.6%
2001 1.0% 0.9% 0.0% 0.0% 5.8% -16.4%
2002 1.8% 1.1% -0.7% 0.0% 7.6% -16.5%
2003 2.8% 2.4% -0.4% 0.0% 7.3% -3.2%
2004 3.8% 4.5% 0.8% 0.0% 8.1% 17.3%
2005 3.3% 4.0% 0.6% 0.0% 12.8% 6.8%
2006 2.7% 3.1% 0.4% 0.0% 2.1% 8.6%
2007 1.8% 3.0% 1.2% 0.0% -2.9% 12.7%
2008 -0.3% -0.6% -0.4% 0.0% -9.2% -17.3%
2009 -2.8% -1.4% 1.3% 0.0% -11.9% -22.5%
2010 2.5% 4.0% 1.5% 0.0% 0.1% 20.3%
2011 1.6% 2.7% 1.1% 0.0% -4.5% 11.4%
2012 2.3% 3.5% 1.2% 0.0% 6.5% 8.7%
2013 2.2% 5.0% 2.8% 0.1% 11.4% 19.1%
2014 2.4% 4.3% 1.9% 0.0% 5.8% 17.5%
What to minimize: Sum E.^2 0.2%
H:AbilitiesProjects2015EconometricsBasics.xlsxOptimization
R. GDP gr. in 2000 = 2% + 0.1(4.1%) +
0.1(7.6%) = 3.2%

Using Excel Solver to run the Optimization
18
What to change
Constant 1.37%
b 1 Home price 0.133
b 2 S&P 500 0.054
Y Y est. Error Error^2 X1 X2
R. GDP Estimate Residual Residual^2
Home
price S&P 500
2000 4.1% 2.3% -1.8% 0.0% 4.1% 7.6%
2001 1.0% 1.3% 0.3% 0.0% 5.8% -16.4%
2002 1.8% 1.5% -0.3% 0.0% 7.6% -16.5%
2003 2.8% 2.2% -0.6% 0.0% 7.3% -3.2%
2004 3.8% 3.4% -0.4% 0.0% 8.1% 17.3%
2005 3.3% 3.4% 0.1% 0.0% 12.8% 6.8%
2006 2.7% 2.1% -0.6% 0.0% 2.1% 8.6%
2007 1.8% 1.7% -0.1% 0.0% -2.9% 12.7%
2008 -0.3% -0.8% -0.5% 0.0% -9.2% -17.3%
2009 -2.8% -1.4% 1.3% 0.0% -11.9% -22.5%
2010 2.5% 2.5% 0.0% 0.0% 0.1% 20.3%
2011 1.6% 1.4% -0.2% 0.0% -4.5% 11.4%
2012 2.3% 2.7% 0.4% 0.0% 6.5% 8.7%
2013 2.2% 3.9% 1.7% 0.0% 11.4% 19.1%
2014 2.4% 3.1% 0.7% 0.0% 5.8% 17.5%

Now let’s run a Regression
19
What to change
Constant 1.37%
b 1 Home price 0.133
b 2 S&P 500 0.054
LINEST () Regression with two ind. Variables
X2 X1
S&P 500 Home pr. Intercept
Coefficient 0.054 0.133 1.37%
SE of Coeff 0.018 0.036 0.003
R Square St. Error 0.757 0.009 #N/A
F Stat df Residual 18.7 12 #N/A
SS Regres SS Residual 0.003 0.1% #N/A
Rearranging LINEST() results in a standard format
Intercept 1.37% 0.003 5.40 0.0002
Home price 0.133 0.036 3.65 0.0033
S&P 500 0.054 0.018 3.05 0.0100
Optimization results w/ Solver
Regression with Data Analysis toolpack
Regression using LINEST ()
Running a regression using the Data Analysis toolpack
or using LINEST () generates the exact same sum of
squared errors and regression coefficients as when
running the optimization with Solver.
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.870
R Square 0.757
Adj. R Square 0.716
St. Error 0.9%
Observations 15
ANOVA
df SS MS F Signif. F
Regression 2 0.003 0.002 18.7 0.0
Residual 12 0.1% 8.19E-05
Total 14 0.004
Intercept 1.37% 0.003 5.40 0.0002
Home price 0.133 0.036 3.65 0.0033
S&P 500 0.054 0.018 3.05 0.0100

Regression = Optimization
20
Both methods do the exact same thing by minimizing the sum of the square of the
Errors or Residuals. Consequently, they generate the exact same overall model with
identical independent variable coefficients.
The big difference is that the standard Regression output generates a lot of
information about the model that Optimization does not do.
Multiple R 0.870
R Square 0.757
Adj. R Square 0.716
St. Error 0.9%
Observations 15
R Square: same meaning as defined within Linear Regression
section.
Adjusted R Square: it adjusts R Square downward for using more
variables. So, Adj. R Square is always a bit smaller than R Square.
Unlike R Square, Adj. R Square can have negative values (for really
bad fitting models).
Standard Error: same meaning as defined within Linear Regression section.
So whenever you can you should use Regression instead of Optimization. But,
Optimization is more flexible, as it can handle constraints on the independent variables
(maybe one of the Xs coeff. should be negative or < 1 for some reason). Regression
can’t handle such constraints.

More Regression info: statistical
significance of independent variables
21
How do we know if variables (Home Price, S&P 500) truly help in explaining and
estimating R. GDP growth)?
The Regression Output tells you whether such variables are statistically significant.
To investigate if Home price chg. is statistically significant, the Regression Output
discloses the Standard Error of that specific regression coefficient: 0.036. Then, it
discloses the t Statistic of this coefficient. It is equal to the regression coefficient/St.
Error: 0.133/0.036 = 3.65. Next, it figures what is the P-value using the t distribution
TDIST(t Stat, df Residual, 2-tail). In this case, it is: TDIST(3.65, 12,2) = 0.0033. This P
value indicates there is only a very small probability that this regression coefficient is
Zero. Thus, we are confident this variable does help explain and estimate R.GDP
growth.

A Visual Summary. Two Independent
Variables for one Model
22-4.0%
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Actual
Estimate
R GDP growthActual vs Estimate
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RGDPgrowth
Home price change
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0.0%
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-30.0% -20.0% -10.0% 0.0% 10.0% 20.0% 30.0%
RGDPgrowth
S&P 500 change
R GDP growthvs S&P 500 change

3) Building an Econometrics Model
23

Do Leading Indicators lead Real GDP growth?
Are they good predictors of Real GDP growth?
• We will build econometric models to address those
questions.
• We will test those models using state-of-the-art peer-
review practices.
24

The Leading Indicators
1 Hours Avg. Weekly Hours - Manufacturing, (Hours)
2 Un_claim Average Weekly Initial Claims - Unemployment Insurance, (Ths.)
3 New_orders Manufacturers' New Orders - Consumer Goods and Materials, (Mil. 1982 $)
4 Nondef1 Manufact. New Orders - Nondefense capital goods exclud. aircraft, (Mil. 1982 $)
5 Nondef2 Manufacturers' New Orders - Nondefense Capital Goods, (Mil. Ch. 1982 $)
6 Building_permits Building Permits for New Private Housing Units, (Ths.)
7 S&P 500 Index of stock prices - 500 common stocks, (1941-43=10, NSA)
8 M2 Money supply - M2, (Bil. 2009 $, NSA)
9 Spread Interest rate spread 10-year Treasury bonds less federal funds, (%, NSA)
10 Expectations Consumer Expectations - from the University of Michigan, (1966Q1=100, NSA)
Original source: Conference Board, BEA, Federal Reserve, BLS. Actual source: Moody's Economy.com
25
2015EconometricsLeading indicators.xlsxLeading indicators
We are using a data set going back to 1982. This will allow us to explore the out-of-
sample issue later on with earlier data prior to 1982.

How to structure the Dependent Variable, Real GDP Growth?
26
Unit root test (nonstationary): Unit root test (nonstationary): Unit root test (nonstationary):
tau Stat Critic. val. Type tau Stat Critic. val. Type tau Stat Critic. val. Type
Dickey-Fuller -1.12 -3.15 with Constant, with Trend Dickey-Fuller -0.83 -3.15 with Constant, with Trend Dickey-Fuller -6.79 -2.58 with Constant, no Trend
Augmented DF -4.74 -2.58 with Constant, no Trend
$-
$2,000
$4,000
$6,000
$8,000
$10,000
$12,000
$14,000
$16,000
$18,000
1982q1
1983q3
1985q1
1986q3
1988q1
1989q3
1991q1
1992q3
1994q1
1995q3
1997q1
1998q3
2000q1
2001q3
2003q1
2004q3
2006q1
2007q3
2009q1
2010q3
2012q1
2013q3
Real GDP in 2009 $mm
8.20
8.40
8.60
8.80
9.00
9.20
9.40
9.60
9.80
1982q1
1983q3
1985q1
1986q3
1988q1
1989q3
1991q1
1992q3
1994q1
1995q3
1997q1
1998q3
2000q1
2001q3
2003q1
2004q3
2006q1
2007q3
2009q1
2010q3
2012q1
2013q3
LN(Real GDP in2009 $)
-10.0%
-8.0%
-6.0%
-4.0%
-2.0%
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
1982q1
1983q3
1985q1
1986q3
1988q1
1989q3
1991q1
1992q3
1994q1
1995q3
1997q1
1998q3
2000q1
2001q3
2003q1
2004q3
2006q1
2007q3
2009q1
2010q3
2012q1
2013q3
Real GDP Growthquarterly % chg.
annualized
A unitroot testtestsif a variable isnonstationary. If itis,the Average
and Variance of the time seriesare unstable acrosssubsectionsof the
data. Here,we can see the Avg.iseverincreasing. The Variance is
mostprobablytoo. Those propertieswill renderall statistical
significance inferencesflawed.
We usedthe Dickey-Fuller(DF) test withaConstant,because the
Average > 0, and a Trendbecause the data clearlytrends. The DF
testconfirmsthisvariable hasa unitrootbecause itstau Stat of -1.12
isnot negative enoughvs.the Critical valueof - 3.15.
Many practitioners believe that taking the log of a level variable
is an effective way to fix this problem. It rarely is. The DF test
suggests this logged variable is even more nonstationary than
the original level variable.
Transformingthe variable intoa% change from one periodtothe next
effectivelyrendersitstationary(mean-reverting). We can see now
that boththe Avg. andVariance are likelytoremainmore stable
across varioustimeframes.
The DF testconfirmsthatisthe case as the tau State of-6.79 ismuch
more negative thanthe Critical value of- 2.58 (fora variable withan
Avg. greaterthan zeroand notrend). In thiscase we also usedthe
AugmentedDFtodouble checkthatthisvariable isstationary. It is.
To avoid unit root issues (nonstationary), we will structure the Leading Indicators
in a similar fashion (% change from one period to the next); except for the Spread
(10 Year Treasury – FF) that is already pretty mean-reverting.
Level: has Unit Root
Not mean-reverting
Nonstationary
LN(Level): has Unit Root
Not mean-reverting
Nonstationary
% Chg: No Unit Root
Mean-reverting
Stationary
2015EconometricsLeading indicators.xlsxVisuals

Selecting independent variables
27
1) Select independent variables that are correlated with the dependent
variable at a statistically significant level.
Correlation stat significance
n 123
St. error 0.09 SQRT(1/(n-1))
a level 0.05 Stat. sign. threshold
Correlation 0.18 St. error x 1.96
Within our data associated with 123
quarterly observations, and using a statistical
significance level of 0.05, this corresponds to
a minimum absolute Correlation of 0.18. For
good measure, let’s round up this minimum
Correlation to 0.20.
2) Select the variable lag (spot, lag 1-, lag 2-, lag 3-, lag 4-quarters) associated
with the highest correlation with the dependent variable.
The independent variables are Leading Indicators. Given that, we expect that
some of the quarterly lags will have the highest correlations.
2015EconometricsEconometric models.xlsxVariable Selection

Correlation with Real GDP Growth
28
Hours Un_claim New_orders Nondef1 Nondef2 Building_permits S&P 500 M2 Spread Expectations
Spot 0.50 -0.60 0.68 0.58 0.37 0.43 0.32 -0.21 0.04 0.16
Lag 1 0.33 -0.47 0.52 0.36 0.33 0.44 0.38 0.01 0.09 0.17
Lag 2 0.25 -0.39 0.33 0.26 0.26 0.39 0.34 0.07 0.12 0.18
Lag 3 0.12 -0.22 0.26 0.04 0.04 0.40 0.28 0.10 0.16 0.18
Lag 4 0.17 -0.22 0.12 0.02 -0.01 0.27 0.15 0.14 0.18 0.17
We can in part answer the first question regarding how much the Leading
Indicators lead economic growth… Apparently, not by much. In six out of the
eight Leading Indicators with statistically significant correlations, the Spot
correlation is the highest.

Selecting the variables
29
Hours Un_claim New_orders Nondef1 Nondef2 Building_permits S&P 500 M2 Spread Expectations
Spot 0.50 -0.60 0.68 0.58 0.37 0.43 0.32 -0.21 0.04 0.16
Lag 1 0.33 -0.47 0.52 0.36 0.33 0.44 0.38 0.01 0.09 0.17
Lag 2 0.25 -0.39 0.33 0.26 0.26 0.39 0.34 0.07 0.12 0.18
Lag 3 0.12 -0.22 0.26 0.04 0.04 0.40 0.28 0.10 0.16 0.18
Lag 4 0.17 -0.22 0.12 0.02 -0.01 0.27 0.15 0.14 0.18 0.17
The highlighted variables have statistically significant correlations
with the dependent variable (Real GDP Growth). And, they have
the highest correlations among the various quarterly lags.

30
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Quarterly change in New Orders
New Orders vs R GDP
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Quarterly change in Hours
Hours vs R GDP
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RGDPgrowth
Quarterly change in Unemployment Claims
Unemployment Claims vs R GDP
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RGDPgrowth
Quarterly change in Nondef Spending1
Nondefense Spending1 vs R GDP
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Quarterly change in Nondef Spending2
Nondefense Spending2 vs R GDP
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RGDPgrowth
Quarterly change in Building Permits Lag1
Building Permits Lag1 vs R GDP
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Quarterly change in S&P 500 Lag1
S&P 500 Lag1 vs R GDP
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RGDPgrowth
Quarterly change in M2
M2 vs R GDP
Scatter Plots illustrating the
relationship between the
independent variables and
the dependent one (R GDP
growth).
EconometricsEconometric models.xlsxPlots

Building the Model manually:
Forward Stepwise Regression, 1st step
31
Correlation with Residual of Step 1
0.00 (0.02) 0.12 (0.26) (0.05) (0.12) (0.07) (0.03)
X
New_orders Hours Un_claim Nondef1 Nondef2 Building_permits Lag1S&P 500 Lag1 M2
5.0% 1.4% -4.1% 6.6% 8.9% 17.2% 8.0% 1.6%
5.0% 1.2% -10.8% 5.2% 2.7% 14.4% 10.2% 0.3%
5.0% 0.4% -7.6% 4.4% 6.0% 1.8% 1.7% 1.1%
1.4% 0.6% -10.2% 5.8% 9.6% -1.6% 0.1% 1.1%
-1.4% -0.2% 5.1% 2.7% 1.4% 8.9% -3.2% 1.2%
0.0% -0.4% 5.4% -0.8% -2.2% -3.6% -2.9% 0.5%
Y X
Real GDP Y est. Residual New_orders
1983q2 9.4% 6.7% -2.8% 5.0%
1983q3 8.1% 6.7% -1.4% 5.0%
1983q4 8.5% 6.7% -1.8% 5.0%
1984q1 8.2% 3.8% -4.4% 1.4%
1984q2 7.2% 1.5% -5.7% -1.4%
1984q3 4.0% 2.7% -1.3% 0.0%
1984q4 3.2% 2.1% -1.1% -0.7%
First step: build a simple linear
regression model with the independent
variable with the highest absolute
correlation with the dependent one. In
this case, it is New_orders with a
correlation of 0.68.
Next, select a 2nd independent
variable with the highest
correlation with the residual of
this first linear regression. As
shown it is Nondef1 with a
correlation of -0.26.
EconometricsEconometric models.xlsxStep1

Forward Stepwise Regression, 2nd step
32
Y Y est X1 X2
Real GDP Estimate Residual New_orders Nondef1
1983q2 9.4% 7.0% -2.5% 5.0% 6.6%
1983q3 8.1% 6.7% -1.4% 5.0% 5.2%
1983q4 8.5% 6.6% -1.9% 5.0% 4.4%
1984q1 8.2% 4.6% -3.6% 1.4% 5.8%
1984q2 7.2% 2.3% -4.9% -1.4% 2.7%
1984q3 4.0% 2.5% -1.5% 0.0% -0.8%
1984q4 3.2% 2.2% -1.0% -0.7% 0.1%
Second step: build a multiple linear
regression model with the two
selected independent variables:
New_orders and Nondef1.
Correlation with Residual of Step 2
0.00 0.00 0.02 0.09 0.13 (0.11) (0.05) (0.11)
X1 X2
New_orders Nondef1 Hours Un_claim Nondef2 Building_permits Lag1S&P 500 Lag1 M2
5.0% 6.6% 1.4% -4.1% 8.9% 17.2% 8.0% 1.6%
5.0% 5.2% 1.2% -10.8% 2.7% 14.4% 10.2% 0.3%
5.0% 4.4% 0.4% -7.6% 6.0% 1.8% 1.7% 1.1%
1.4% 5.8% 0.6% -10.2% 9.6% -1.6% 0.1% 1.1%
-1.4% 2.7% -0.2% 5.1% 1.4% 8.9% -3.2% 1.2%
0.0% -0.8% -0.4% 5.4% -2.2% -3.6% -2.9% 0.5%
-0.7% 0.1% -0.2% 4.7% -1.8% -12.6% 3.1% 1.6%
Next, select a 3d independent
variable with the highest
correlation with the residual of
this second regression. As
shown it is Nondef2 with a
correlation of 0.13. This
correlation is probably too low.
We suspect this variable will not be adequately statistically significant when included in
the model. Let’s check…

Forward Stepwise Regression, 3d step
33
Coefficients St. Error t Stat
Intercept 2.6% 0.2% 15.8
X1 New_orders 0.643 0.091 7.1
X2 Nondef1 0.286 0.069 4.2
X3 Nondef2 -0.087 0.041 (2.1)
Actually, the issue with X3 Nondef2 is not
that it is not statistically significant, but
that its regression coefficient has the
wrong sign relative to its original
correlation with the dependent variable.
That’s a concern.
Let’s redo this 3d step with the next independent variable that had the 2nd highest
absolute correlation with the residual from the regression in the 2nd step. It was
Building_permits Lag 1 with a correlation of (0.11). That correlation appears too low.
We suspect again this variable will not be adequately statistically significant. But, let’s
give it a try to find out…

Forward Stepwise Regression, 3d step, 2nd try
34
Coefficients St. Error t Stat P-value
Intercept 2.6% 0.2% 15.7 0.00
X1 New_orders 0.556 0.100 5.6 0.00
X2 Nondef1 0.192 0.054 3.5 0.00
X3 Building_permits Lag10.040 0.028 1.4 0.16
Actually, Building_permits did better
than expected. t Stat of 1.4 and P-
value of 0.16 can be deemed
acceptable if the variable and its
regression coefficient sign make good
sense; which in this case they do.
EconometricsEconometric models.xlsxStep3b
Given the already very low correlation coefficients associated with this 3d regression it
is not worth going on to a 4th regression to select a 4th independent variable. So, our
model will at most have three independent variables.
The next step is to check if adding this 3d independent variable is even worth it? Does
it add much incremental information over the model with just two independent
variable?

Comparing model with two vs. three
independent variables
35
Hold out performance
2 var 3 var
Actual Model Model
2014q1 -2.1% 2.2% 2.5%
2014q2 4.6% 4.0% 3.7%
2014q3 5.0% 3.9% 4.0%
2014q4 2.6% 1.7% 1.8%
2014 2.5% 2.9% 3.0%
Regression Stats 2 var. 3 var.
Multiple R 0.718 0.724
R Square 0.516 0.524
Adj. R Square 0.508 0.512
Standard Error 1.83% 1.83%
Observations 123 123
Regression coefficients. 2 variable model
Intercept 2.6% 0.2% 15.5 0.00
New_orders 0.616 0.091 6.8 0.00
Nondef1 0.197 0.054 3.6 0.00
Regression coefficients. 3 variable model
Intercept 2.6% 0.2% 15.7 0.00
New_orders 0.556 0.100 5.6 0.00
Nondef1 0.192 0.054 3.5 0.00
Building_permits Lag10.040 0.028 1.4 0.16
The two models are just about even on
Goodness-of-fit measures.
In the two-variable model both variables are
very statistically significant.
In the three-variable model, the 3d one, as
mentioned is not statistically significant.
In the Hold Out, the 2- var. model performs
just as well if not better than the 3-var. one.
All of the above suggests the 2-var. model is
the winner as the 3d variable does not add
enough incremental information.
EconometricsEconometric models.xlsxCompare 2 vs 3b

Model with 2 variables. Variables’ Influence
36
R² = 0.4635
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RealGDPgrowth
New_orders
New_orders vs Real GDP growth
R² = 0.3321
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Nondef1
Nondef1 vs Real GDP growth
R² = 0.5161
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2 var. model estimate
2 variable model vs Real GDP growth
EconometricsEconometric models.xlsxReg Model testing.xlsxMulticollinearity
New_orders
has a stronger
influence on
the fit of the
model.

Historical fit & Error Reduction
37
Real GDP growth
Average 2.9%
St. Deviation 2.62%
St. Error 1.83%
Error reduction -29.9%
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1983q2
1984q4
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1992q2
1993q4
1995q2
1996q4
1998q2
1999q4
2001q2
2002q4
2004q2
2005q4
2007q2
2008q4
2010q2
2011q4
2013q2
2014q4
Real GDP growthvs. 2 var model estimate
Real GDP Estimate
Average 2.9%
EconometricsEconometric models.xlsxReg Model testing.xlsxMulticollinearity
This is a very simple, yet powerful
way to assess the effectiveness of a
model. In the absence of any model,
you could simply use the historical
average economic growth of 2.9% as
a forecast. In essence, you would
accept the Standard Deviation of this
variable of 2.62% as your model’s
Standard Error. This is sometimes
called a Naïve model.
Next, you check how much lower is the Standard Error of your actual model vs. the Standard
Deviation of the variable: (1.83%)/2.62% = -29.9%. That’s not bad…

Adding an Autoregressive Variable (Y Lag 4)
38
Model
Model 2-var
2-var + Y Lag 4
Regression Stats
Multiple R 0.718 0.750
R Square 0.516 0.563
Adj. R Square 0.508 0.552
St. Error 1.83% 1.75%
Observations 123 123
Coefficient
Intercept 2.6% 2.0%
New_orders 0.62 0.65
Nondef1 0.20 0.18
Y Lag 4 0.21
Standardized coefficient
Nondef1 0.28 0.25
Y Lag 4 0.22
T Stat
Intercept 15.5 8.4
New_orders 6.8 7.4
Nondef1 3.6 3.4
Y Lag 4 3.6
P value
Intercept 0.00 0.00
Nondef1 0.00 0.00
Y Lag 4 0.00
Model
Model 2-var
Actual 2-var + Y Lag 4
2014q1 -2.1% 2.2% 2.1%
2014q2 4.6% 4.0% 3.7%
2014q3 5.0% 3.9% 4.2%
2014q4 2.6% 1.7% 1.8%
2014 2.5% 2.9% 3.0%
If you know economic growth 4 quarters ago, it does
provide marginally additional incremental info on
estimating economic growth in current quarter.
Coefficients for New_orders and Nondef1 have
remained surprisingly stable and so have their
influence as measured with Standardized
coefficients.
Statistical
significance of
variables is very
similar for both
models.
Hold Out
performance
is pretty
much even
EconometricsEconometric models.xlsxModel finalists

Visual comp.: Regression vs Autoregressive model
39
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QuarterlyRealGDPchange,annualized
Reg model:Actual vs Estimate
Actual Estimate
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QuarterlyRealGDPchange,annualized
Autoreg model:Actual vs Estimate
Actual Estimate
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2014q1 2014q2 2014q3 2014q4 Average
Hold Out Performance
Actual Reg est. Autoreg est.
0.00
0.10
0.20
0.30
0.40
0.50
0.60
New_orders Nondef1 Y Lag 4
#ofStandarddeviations
StandardizedRegressioncoefficient
Reg Autoreg
Econometric models.xlsxGraphs.xlsxComparison

The Pros & Cons of Autoregressive Models
40
The pros:
1) It often reduce the autocorrelation of residuals;
2) It improves the overall Goodness-of-fit of a model;
3) It often improves the forecasting up to the Lag used in the model (Lag 4
quarters will allow you to forecast potentially better up to 4 quarters out).
The cons:
1) The autoregressive variable can grab away explanatory information from the
macroeconomic variables and weaken their statistical significance;
2) It can weaken the forecasting beyond the Lag used in the model. If you use
Lag 4 quarters, the model forecasting may weaken beyond 4 quarters.
Thus, depending on what is your objective and the issues associated with a model, an
autoregressive model may add value or not. You may decide to keep both models and
use them in different circumstances.
In this specific example, the autoregressive model does not add much value.

4) Model Testing
41
Linear Regression underlying assumptions:
1) No near-exact linear relationships between independent variables. Multicollinearity issue.
2) Error terms (Residuals) are independent. Autocorrelation issue.
3) Residuals have a constant variance. Heteroskedasticity issue.
We will test the regular Regression model with two variables for all of the above
assumptions, and conduct additional tests related to model specification.

Multicollinearity
42
To test an independent variable for
multicollinearity, you run a
regression using it as a dep. variable
and use all other ind. variables to
regress it. If that model’s resulting
RSquare > 0.75, you may have a
multicollinearity issue.
The literature focuses on the Variance Inflation Factor (VIF). But, SQRT(VIF) is more
interesting as it denotes the coefficient’s Standard Error multiple. So, if VIF is 4, SQRT(VIF) is 2
and the coefficient’s Standard Error is 2 x as large and the t Stat half of what it would be if
multicollinearity was not an issue. (Source: John Fox 1991). A short cut to calculating
SQRT(VIF) is to run a model with only the one variable being tested. And, divide the Standard
Error of this variable’s coeff. within the multiple regression model by the one within the linear
regression (with only that one variable). And, you get SQRT(VIF). (EViews documentation).
In R, you can calculate the VIF using the vif( ) function with the car package.
The two variables have the same exact VIF
because they are regressed against each other
without any additional variables.
EconometricsEconometric models.xlsxReg model testing.xlsxmulticollinearity
Regressing New_orders (Y), using Nondef1 (X)
Multicollinearity test
Threshold
Actual Severe Conservative Standard
R 0.57 0.87 0.89 0.95
RSquare 0.32 0.75 0.80 0.90
1 - R Squ. Tolerance 0.68 0.25 0.20 0.10
1/Tolerance VIF 1.48 4 5 10
SQRT(VIF) 1.22 2.0 2.2 3.2

2-variable Model Residuals
43
Econometric models.xlsxReg model testing.xlsxautocorrelation
Unless the residual pattern is extremely obvious, it is difficult to visually accurately
assess whether residuals are autocorrelated or heteroskedastic. You have to statistically
test for those properties to get an accurate diagnostic. However, we can speculate that
the residuals are probably not very heteroskedastic (right hand side scatter plot).
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Residual
Estimate
2-var model:Residual l vs Estimate
Econometric models.xlsxGraphs.xlsxRegression model

Autocorrelation Lag 1 test: Durbin Watson (DW)
44
In R with lmtest package.
> dwtest(Regression, order.by = NULL, exact = NULL)
Durbin-Watson test
data: Regression
DW = 1.7189, p-value = 0.05316
alternative hypothesis: true autocorrelation is greater than 0
P-value Interpretation
0.05 We can reject alternative hypothesis that true autocorrelation is >0.
0.95 We can't reject the alternative hypothesis that the true autocorrelation is >0.
Durbin Watson
Numerator 6.94% sum(Residual - Residual t-1)^2
Denominator 4.04% sum(Residual^2)
DW score 1.719
n 123
k 2
dL 1.634
dU 1.715
Value from DW table
Value from DW table
number of observations
number of independent variables
The 1.719 DW score falls just outside
the zone of uncertainty for positive
autocorrelation (1.634 – 1.715). So,
we can be pretty sure those residuals
are not positively autocorrelated with
Lag 1 residuals.
The R output says the same thing.
There is only a 0.05 chance that such
residuals are autocorrelated. In R,
watch for the direction of this test.
EconometricsEconometric models.xlsxReg model testing.xlsxautocorrelation

Two better tests than DW: Ljung-Box & Breusch-Godfrey
45
Comparing Ljung-Box and Breusch-Godfrey tests using R
Ljung-Box test Breusch-Godfrey test.
You don't need to load any extra library for this test. In R with lmtest package.
Testing for Lag 1 or AR(1) Testing for Lag 1 or AR(1)
> bgtest(Regression, order = 1, type = c("Chisq"))
Breusch-Godfrey test for serial correlation of order up to 1
data: Regression
LM test = 2.2148, df = 1, p-value = 0.1367
Testing up to Lag 4 or AR(4) Testing up to Lag 4 or AR(4)
> Box.test(Regression$res,lag = 4, type = c("Ljung-Box"),fitdf = 0) > bgtest(Regression, order = 4, type = c("Chisq"))
Box-Ljung test Breusch-Godfrey test for serial correlation of order up to 4
data: Regression$res data: Regression
X-squared = 37.1735, df = 4, p-value = 1.659e-07 LM test = 24.7465, df = 4, p-value = 5.657e-05
2015EconometricsReg Model testing.xlsxAutocorrelation
Autocorrelation related p-value
LB BG Interpretation
AR(1) 0.1355 0.1367 Not stat. significant
AR(4) 0.0000 0.0000 Very stat. significant
The LB and BG tests are better than DW for
two reasons. They can test for more than one
lag. They can also test a model with an
autoregressive variable. Meanwhile, DW
can’t.
The LB and BG tests diagnostics were nearly
identical. Residuals do not have an AR(1)
process. But, they have an AR(4) one.

Autocorrelations statistical significance
46
Autocorrelation tests
Correl. SE t stat P value
Lag 1 0.13 0.09 1.48 0.14
Lag 2 0.42 0.09 4.66 0.00
Lag 3 0.21 0.09 2.34 0.02
Lag 4 0.25 0.09 2.77 0.01
Notice that the P value for Lag 1 is very close to the P value for the Ljung-Box and
Breusch-Godfrey tests shown on previous slide. All three test approaches seem more
sensitive than DW that came up with a very low P value that Residuals would be
autocorrelated.

Autocorrelation: Regular model vs Autoregressive one
47
Regular model
Autocorrelation statistical significance
Lag 1 0.13 0.09 1.48 0.14
Lag 2 0.42 0.09 4.66 0.00
Lag 3 0.21 0.09 2.34 0.02
Lag 4 0.25 0.09 2.77 0.01
Autoregressive Model
Autocorrelation statistical significance
Lag 1 0.01 0.09 0.13 0.90
Lag 2 0.27 0.09 2.95 0.00
Lag 3 0.10 0.09 1.08 0.28
Lag 4 0.02 0.09 0.20 0.84
By adding a Y Lag 4 variable, the Autoregressive model reduced all autocorrelations
(from Lag 1 to Lag 4) vs. the Regular model. This is a common phenomenon in
modeling. Notice the Autoregressive Model would not entirely circumvent the
autocorrelation of residual issue. The Lag 2 is clearly statistically significant.

Heteroskedasticity test: Breusch-Pagan
48
Y X1 X2
Residual^2 New_ordersNondef1
1983q2 0.1% 5.0% 6.6%
1983q3 0.0% 5.0% 5.2%
1983q4 0.0% 5.0% 4.4%
1984q1 0.1% 1.4% 5.8%
1984q2 0.2% -1.4% 2.7%
Breusch-Pagan LM Chi dist. P value
Lagrange Multiplier (LM) 0.8
DF (# variables) 2.0
Chi Dist. P value 0.68
Multiple R 0.079
R Square 0.006
Adj. R Square -0.010
Standard Error 0.000
Observations 123
ANOVA
Regression 2 1.75E-07 8.77E-08 0.38 0.68
Residual 120 2.77E-05 2.31E-07
Total 122 2.78E-05
In R with lmtest package.
> bptest(Regression,varformula = NULL, studentize = FALSE)
Breusch-Pagan test
data: Regression
BP = 0.8135, df = 2, p-value = 0.6658
EconometricsEconometric models.xlsxReg model testing.xlsxheteroskedasticity
The BP test tests for linear heteroskedasticity. It
suggests that residuals are not heteroskedastic
because the LM Chi distribution P value at 0.68 is far
from being statistically significant. In most cases, the
ANOVA F test generates very similar values.

Heteroskedasticity test: White Test
49
Y
Residual^2 X1 X2 X1^2 X2^2
1983q2 0.1% 5.0% 6.6% 0.2% 0.4%
1983q3 0.0% 5.0% 5.2% 0.2% 0.3%
1983q4 0.0% 5.0% 4.4% 0.3% 0.2%
1984q1 0.1% 1.4% 5.8% 0.0% 0.3%
1984q2 0.2% -1.4% 2.7% 0.0% 0.1%
ANOVA
df SS MS F Signific. F
Regression 4 5.08E-07 1.27E-07 0.55 0.70
Residual 118 2.73E-05 2.32E-07
Total 122 2.78E-05
White Test LM Chi dist. P value
Lagrange Multiplier (LM)2.2
DF (# variables) 4.0
The White Test tests for linear and
nonlinear heteroskedasticity. You can see
how its regression is specified with all the
2nd degree variables. This test confirms
that residuals are not heteroskedastic
even on a nonlinear basis.

Heteroskedasticity test: Autoregressive
Conditional Heteroskedasticity (ARCH)
50
Y X1 X2 X3 X4
Resid^2 Resid^2 t-1 Resid^2 t-2 Resid^2 t-3 Resid^2 t-4
1983q2 0.1%
1983q3 0.0% 0.1%
1983q4 0.0% 0.0% 0.1%
1984q1 0.1% 0.0% 0.0% 0.1%
1984q2 0.2% 0.1% 0.0% 0.0% 0.1%
1984q3 0.0% 0.2% 0.1% 0.0% 0.0%
1984q4 0.0% 0.0% 0.2% 0.1% 0.0%
ANOVA
df SS MS F Sign. F
Regression 4 5.21E-07 1.3E-07 0.56 0.69
Residual 114 2.63E-05 2.31E-07
Total 118 2.68E-05
ARCH LM Chi dist. P value
Lagrange Multiplier (LM) 2.3
DF (# lags) 4
This heteroskedasticity test checks
whether the variance of an error
term is a function of the size of the
previous error terms.
In plain English, are large residuals
followed by large residuals and
small ones by small ones.
As indicated with the high value for
Significance of F and Chi
distribution P value, this model’s
residuals do not suffer from this
type of heteroskedasticity.

Where does heteroskedasticity come from?
51
-6.0%
-4.0%
-2.0%
0.0%
2.0%
4.0%
6.0%
-8.0% -6.0% -4.0% -2.0% 0.0% 2.0% 4.0% 6.0% 8.0%
Residual
Estimate
Reg model:Residual l vs Estimate
-6.0%
-4.0%
-2.0%
0.0%
2.0%
4.0%
6.0%
-10.0% -8.0% -6.0% -4.0% -2.0% 0.0% 2.0% 4.0% 6.0%
Residual
New_orders
Reg model:Residual vs New_orders
-6.0%
-4.0%
-2.0%
0.0%
2.0%
4.0%
6.0%
-16.0% -11.0% -6.0% -1.0% 4.0% 9.0%
Residual
Nondef1
Reg model:Residual vs Nondef1
Econometric models.xlsxGraphs.xlsxRegression model
We already know the overall model does not demonstrate heteroskedastic residuals. But
if it is some model reviewers fit a quadratic regression line to the residuals vs. each of the
independent variables to identify heteroskedasticity at the variable level. In this case, the
resulting quadratic regression lines are pretty flat reflecting unlikely heteroskedasticity
issues.

Testing Heteroskedasticity at the
variable level: Park Test
52
ANOVA
Regression 1 0.000 0.000 0.233 0.63
Residual 121 0.000 0.000
Total 122 0.000
Coeff. St. Error t Stat P-value
Intercept 0.00 0.00 7.42 0.00
New_orders 0.00 0.00 0.48 0.63
EconometricsEconometric models.xlsxReg model testing.xlsxheteroskedasticity variable
The most common form of the Park test is to log all the
variables. But, you can’t log negative values. So, the
Park test has also a linear form described here. Notice
that this version of the Park test is nearly identical to
the Breusch-Pagan test except it tests for one single
variable at a time to identify where the
heteroskedasticity comes from.
Using linear form of the Park test.
Y X1
Residual^2 New_orders
1983q2 0.06% 5.0%
1983q3 0.02% 5.0%
1983q4 0.04% 5.0%
1984q1 0.13% 1.4%
We ran the same test for Nondef1, and got P
values of 0.38. So, in both cases the
residuals are not heteroskedastic relative to
the level of either independent variables.

Residual autocorrelation &
heteroskedasticity recap
53
Residuals are not heteroskedastic. Note that all the
heteroskedasticity tests (BP, White, ARCH) for the overall model
generated almost the same Sign. of F and Chi Square dist. P-
value (all near 0.7). That’s even though they tested for different
shapes of heteroskedasticity.
Residuals are autocorrelated when looking beyond Lag 1. There
too a couple of the tests (Ljung-Box, Breusch-Pagan) gave us
nearly identical results in terms of respective P values.
There are several ways to resolve autocorrelation and
heteroskedasticity issues as shown on the next slide.

How to resolve Autocorrelation & Heteroskedasticity
54
C:UsersliongcDesktopEconometricsModel GuidanceModel guidance map.xlsxSimple Map
A Unit Root issue often leaves a footprint in residual issues such as autocorrelation and
heteroskedasticity. The above diagram shows the three ways to resolve autocorrelation
and heteroskedasticity issues.
Calculate RobustStandardError: Newey-West.
Recalculate ind.variablestatistical significance.
Introduce anautoregressive variable:Y Lag 4.
Feasible GeneralizedLeastSquaresModel (FGLS)
Calculate RobustStandardError: White.
Recalculate ind.variablestatistical significance.
TransformY variable. Detrendmore if possible.
WeightedLeastSquaresModel (WLS)
Autocorrelation
Heteroskedasticity
UnitRoot
Stationarity issue

Mapping the Robust Standard Error Path
55
C:UsersliongcDesktopEconometricsModel GuidanceModel guidance map.xlsxMap 2
The diagram below fleshes
out what it means to
calculate Robust Standard
Error and test ind. variables
statistical significance.
Yes
No Yes
No
Yes
No
Are residuals
heteroskedastic(White
test,Breusch-Pagan)
and/orautocorrelated
(DW,Ljung)?
Calculate RobustStandard
Error: White for heteros.;
Newey-Westforautocor.
Recalculate independent
variablesstatistical
significance.
Are
independent
variablesstill
statistically
significant?
Good,you are done with
heteroskedasticityand
autocorrelationtesting.
Good,you are done with
resolving
heteroskedasticityand
autocorrelationissues.
Confirmthata variable
that isnot stat.
significantissupported
by economictheory. If
not,considerremoving
variable.
Is yourdependent
variable stationary
[UnitRoot ] (Dickey-
Fullertest)?
Consider
transformingY
to % change or
First-Difference .

White SEs = Newey-West SEs w/ zero lag with R
56
Testing that White SE (hc1) = Newey-West (0 Lag, alt.model, small sample adjustment).
For the Regression model.
> library(car)
> sqrt(diag(hccm(Regression,type=c("hc1"))))
(Intercept) New_orders Nondef1
0.001654879 0.101647319 0.056301194
> library(sandwich)
> sqrt(diag(NeweyWest(Regression,lag=0,prewhite=FALSE,adjust=123/121)))
(Intercept) New_orders Nondef1
0.001654879 0.101647319 0.056301194
This calculates the original
White SE “HC1” version
developed by White but
adjusted for small sample.
This is the Newey-West SE with a manual adjustment for small sample. The adjustment is
described as: n/(n – k).
n is sample size (123)
k is number of parameters.
The literature describes “k” as including the intercept and “n – k” being equal to the df of
residual for the regression (n – k = 120). But, I got exact results White SEs = Newey-West
SEs by treating k as number of independent variables excluding the intercept with n – k =
121.
EconometricsEconometric models.xlsxReg model testing.xlsxStat sign.
Another way to get White SEs = N-W SEs is to use the White SEs “hc0” version that
excludes a small sample adjustment. When doing N-W, you would not enter the small
sample adj. argument. If you have a very large sample using those would be fine.

White SE = Newey-West SE summary
57
White hc1 = Newey-West (with zero lag) with small sample adjustment
White hc0 = Newey-West (with zero lag) without small sample adjustment
If sample is very large, the sample adjustment may be immaterial.
Using White hc2 or White hc3 will result in Robust SEs adjusted for
heteroskedasticity that will often be much larger than Robust SEs adjusted for
both heteroskedasticity and autocorrelation (N-W), a rather incoherent
outcome.

Recalculating variables stat. significance
58
The N-W SE with up to a 4 quarter lag is
higher for the Nondef1 but is actually
lower for New_orders vs the N-W SE with
0 lag or the White SE. This is not a typo (I
reran R several times to double check it).
EconometricsEconometric models.xlsxReg model testing.xlsxStat sign.
Existing Stat. Sign. with regular Standard Errors
Coeffic.
Standard
Error t Stat P-value
New_orders 0.616 0.091 6.75 0.00
Nondef1 0.197 0.054 3.61 0.00
Recalculating Stat. Sign. with White S.Es
Coeffic. White SE t Stat P-value
New_orders 0.616 0.102 6.06 0.00
Nondef1 0.197 0.056 3.49 0.00
The t Stat = Regression coefficient/Robust Standard Error
P-value = TDIST(abs(t Stat), DF of Residual, 2)
Recalculating Stat. Sign. with Newey-West S.Es. Lag=4
Coeffic. N-W SE t Stat P-value
New_orders 0.616 0.097 6.33 0.00
Nondef1 0.197 0.059 3.36 0.00
The t Stat = Regression coefficient/Robust Standard Error
P-value = TDIST(abs(t Stat), DF of Residual, 2)
Recalculating the stat. significance of
variables with Robust SEs did not have a
material impact. The variables remained
very statistically significant.

Testing coefficient
stability across different
time series
59
Regression Coefficient X2 X1
Nondef1 New_ordersIntercept
Model from 1983q2 to 2013q4 0.197 0.616 2.6%
Model from 1983q2 to 2007q4 0.221 0.604 2.9%
Model from 1985q2 to 2009q4 0.176 0.646 2.6%
Model from 1987q2 to 2011q4 0.193 0.588 2.4%
Model from 1989q2 to 2013q4 0.223 0.523 2.3%
t Stat X2 X1
Nondef1 New_orders
Model from 1983q2 to 2013q4 3.6 6.8
Model from 1983q2 to 2007q4 3.9 5.8
Model from 1985q2 to 2009q4 3.2 6.9
Model from 1987q2 to 2011q4 3.3 6.1
Model from 1989q2 to 2013q4 3.6 5.2
P value X2 X1
Nondef1 New_orders
Model from 1983q2 to 2013q4 0.00 0.00
Model from 1983q2 to 2007q4 0.00 0.00
Model from 1985q2 to 2009q4 0.00 0.00
Model from 1987q2 to 2011q4 0.00 0.00
Model from 1989q2 to 2013q4 0.00 0.00
Goodness-of-fit of models
R Square St. Error
Model from 1983q2 to 2013q4 0.516 1.83%
Model from 1983q2 to 2007q4 0.446 1.70%
Model from 1985q2 to 2009q4 0.562 1.71%
Model from 1987q2 to 2011q4 0.530 1.78%
Model from 1989q2 to 2013q4 0.524 1.75%
We reran this regression by using four
different periods of 14.5 years each
every two years to observe how
stable the regression coefficients are.
Regression coefficients
are overall pretty stable.
Statistical significance of
regression coefficients held
up well for all regressions.
Goodness-of-fit of the various models as
measured by R Square and Standard Error
remained reasonably stable too.
EconometricsEconometric models.xlsxReg model testing.xlsxCoefficient stability

Exploring Outliers (why R can be really cool)
60
> influencePlot(Regression, id.n=6) Cook's D
(bubble size)
It measures the change to
the estimates that results
from deleting an
observation. Its
calculation combines a
measure of Outlierness
(like Stud. Residuals) and
Leverage.
Threshold:> 4/n
Studentized Residuals
(y-axis)
Dependent variable
outliers
Large error. Unusual
dependent variable value
given the independent
variable’s input. This means
an actual datapoint is two
standard deviations (scaled
on a t distribution) of the
Residual away from the
regressed line. Fairly similar
to being two model's
Standard Errors away (small
differences due to dfs).
Threshold: + or - 2.
Hat-Leverage
(x-axis)
Independent variable outliers
Leverage measures how far an independent
variable deviates from its Mean. Threshold:
>(2k + 2)/n
EconometricsGraphs.xlsxR Graphs
Using the car package

Outliers
61
> influencePlot(Regression, id.n=6)
StudRes Hat CookD
5 2.81192728 0.026683421 0.261387671
9 1.31478653 0.060466331 0.191990916
23 1.43797317 0.068965131 0.224956383
33 -0.79884290 0.066833211 0.123615536
69 2.63048375 0.038623597 0.297165815
100 -2.14789027 0.029868097 0.214386600
102 -0.01900992 0.070398888 0.003032988
103 -1.58568051 0.176556380 0.421266201
104 0.30230346 0.199567139 0.087481253
106 -2.44084923 0.025118592 0.221672137
112 -2.57586760 0.008165951 0.131880907
119 -2.14938212 0.011899339 0.134171528
2 var Regression Model
A B A x B Rank Rank
Observ. StudRes Hat-Lev. CookD Influence CookD Influence
103 -1.586 0.177 0.421 0.280 1 1
69 2.630 0.039 0.297 0.102 2 2
5 2.812 0.027 0.261 0.075 3 5
23 1.438 0.069 0.225 0.099 4 3
106 -2.441 0.025 0.222 0.061 5 7
100 -2.148 0.030 0.214 0.064 6 6
9 1.315 0.060 0.192 0.080 7 4
119 -2.149 0.012 0.134 0.026 8 10
112 -2.576 0.008 0.132 0.021 9 11
33 -0.799 0.067 0.124 0.053 10 9
104 0.302 0.200 0.087 0.060 11 8
102 -0.019 0.070 0.003 0.001 12 12
Correlation 0.87
EconometricsGraphs.xlsxR Graphs
The most important and encompassing outlier-measure is Cook’s D because it
pretty much aggregates the information from Studentized Residuals and Leverage.

Impact of Outliers on regression coefficients
62
> Regression103<-lm(Real.GDP~New_orders + Nondef1, econdata,subset=-c(103,124,125,126,127))
> summary(Regression103)
Regression testing for outliers
without without without
All data 103 69 5
Coeffic.
Intercept 0.026 0.027 0.026 0.026
New_orders0.616 0.575 0.648 0.647
Nondef1 0.197 0.186 0.173 0.179
t Stat
New_orders 6.75 6.11 7.21 7.24
Nondef1 3.61 3.40 3.20 3.36
Adj. R Sq. 0.508 0.433 0.522 0.528
EconometricsEconometric models.xlsxReg model testing.xlsxOutliers
Here we reran the regression by taking out one
at a time each of the top three observations
ranked by Cook’s D measure (the more
encompassing measure of influence).
As shown, the coefficients and their statistical
respective statistical significance remained
pretty stable.

Does Cook’s D really work?
63
Should we be concerned about datapoint 104. It
has the highest Leverage combined with a very low
Residual. Hypothesis: could this mean it actually
has a greater influence on regression coefficients
than datapoint 103 that has a pretty large residual?
Regression testing for outliers
Change Change
without without without without
All data 103 104 103 104
Coeffic.
Intercept 0.026 0.027 0.026 1.8% -0.4%
New_orders0.616 0.575 0.621 -6.6% 0.9%
Nondef1 0.197 0.186 0.201 -5.7% 2.1%
t Stat
New_orders 6.75 6.11 6.65
Nondef1 3.61 3.40 3.56
Adj. R Sq. 0.508 0.433 0.463 -14.7% -8.9%
As shown, Cook’s D did work just fine.
Datapoint 103 (large bubble) has much more
influence on the regression coefficient than
datapoint 104 (small bubble).
EconometricsEconometric models.xlsxReg model testing.xlsxOutliers

Are Residuals Normally distributed?
64
Jarque- Bera test.
Probability distribution is Normal
n - k 121
Skewness 0.0
Kurtosis 0.2
JB score 0.1
DF 2
p-value 0.94
EconometricsEconometric models.xlsxReg model testing.xlsxNormality
> qqPlot(Regression)
> hist(rstudent(Regression))
EconometricsEconometric models.xlsxGraphs.xlsxR Graph
Visually by either looking at a QQ Plot or a
histogram, we can see that the Residuals look
pretty normally distributed. Note, the QQ Plot
also describes a 95% CI relative to a Normal
distribution. Also, the Jarque-Bera test confirms
that the Residuals are normally distributed (p
value 0.94).
Need packages: tseries & quadprog
> jarque.bera.test(Regression$res)
Jarque Bera Test
data: Regression$res
X-squared = 0.0529, df = 2, p-value = 0.9739

Scenario Testing: Can the Model break down?
65
EconometricsEconometric models.xlsxReg model testing.xlsxScenario testing
If we use inputs
(yellow), we get
output (pink).
Scenarios: Real GDP quarterly change annualized
New_orders
Min Median Max
3.1% -11.4% -7.3% -3.3% 0.7% 3.1% 5.5% 7.9%
Min -20.6% -8.4% -5.9% -3.5% -1.0% 0.5% 1.9% 3.4%
-13.4% -7.0% -4.5% -2.1% 0.4% 1.9% 3.4% 4.8%
-6.2% -5.6% -3.1% -0.7% 1.8% 3.3% 4.8% 6.2%
Nondef1 Median 0.9% -4.2% -1.7% 0.8% 3.2% 4.7% 6.2% 7.6%
6.3% -3.1% -0.7% 1.8% 4.3% 5.8% 7.2% 8.7%
11.7% -2.1% 0.4% 2.9% 5.4% 6.8% 8.3% 9.8%
Max 17.1% -1.0% 1.5% 3.9% 6.4% 7.9% 9.4% 10.8%
Regression Model Model data from 1982 From beginning of series in 1959
Coefficient Min Median Max Min Median Max
Intercept 2.6%
New_orders 0.616 0.5% -9.3% 0.5% 5.2% -11.4% 0.7% 7.9%
Nondef1 0.197 0.9% -15.7% 0.9% 7.5% -20.6% 0.9% 17.1%
Output estimate
Real GDP 3.1%
Median R GDP
Learning sample 3.1%
Since 1947 3.2%
We then sensitize the
values of both
New_orders and
Nondef1 based on
historical ranges
going back to 1959.
We then generate 49
different scenarios of
GDP growth.

Are the scenario estimates reasonable?
66
Scenarios: Real GDP quarterly change annualized
New_orders
Min Median Max
3.1% -11.4% -7.3% -3.3% 0.7% 3.1% 5.5% 7.9%
Min -20.6% -8.4% -5.9% -3.5% -1.0% 0.5% 1.9% 3.4%
-13.4% -7.0% -4.5% -2.1% 0.4% 1.9% 3.4% 4.8%
-6.2% -5.6% -3.1% -0.7% 1.8% 3.3% 4.8% 6.2%
Nondef1 Median 0.9% -4.2% -1.7% 0.8% 3.2% 4.7% 6.2% 7.6%
6.3% -3.1% -0.7% 1.8% 4.3% 5.8% 7.2% 8.7%
11.7% -2.1% 0.4% 2.9% 5.4% 6.8% 8.3% 9.8%
Max 17.1% -1.0% 1.5% 3.9% 6.4% 7.9% 9.4% 10.8%
Percentiles vs Real GDP history going back to 1947Q2
New_orders
Min Median Max
-11.4% -7.3% -3.3% 0.7% 3.1% 5.5% 7.9%
Min -20.6% 0.003 0.014 0.049 0.121 0.189 0.317 0.535
-13.4% 0.009 0.033 0.075 0.183 0.313 0.531 0.718
Nondef1 -6.2% 0.017 0.057 0.132 0.310 0.525 0.714 0.792
Median 0.9% 0.036 0.089 0.201 0.521 0.708 0.791 0.874
6.3% 0.057 0.131 0.310 0.652 0.785 0.856 0.930
11.7% 0.074 0.180 0.442 0.748 0.831 0.909 0.959
Max 17.1% 0.120 0.276 0.617 0.801 0.887 0.941 0.976
EconometricsEconometric models.xlsxReg model testing.xlsxScenario testing
Some of the
scenarios input may
not be reasonable
because New_orders
and Nondef1 are
positively correlated
(R = 0.52). But, the
resulting output of
the R GDP estimates
percentiles vs entire
series going back to
1947 seems pretty
reasonable. Thus,
the Model does not
appear to break
down readily even
with out-of-sample
variable inputs.
Red = < 10th percentile. Green > 90th percentile

Is the Model well specified? Link test
67
Y
R GDP Y est. Y est.^2
1983q2 9.4% 7.0% 0.5%
1983q3 8.1% 6.7% 0.4%
1983q4 8.5% 6.6% 0.4%
1984q1 8.2% 4.6% 0.2%
1984q2 7.2% 2.3% 0.1%
1984q3 4.0% 2.5% 0.1%
1984q4 3.2% 2.2% 0.0%
1985q1 4.0% 4.1% 0.2%
1985q2 3.7% 1.4% 0.0%
Intercept 0.00 0.003 0.24 0.810
Y est. 1.07 0.106 10.10 0.000
Y est.^2 -2.21 1.961 -1.13 0.261
The Link test checks if your regression is properly specified. If it is one should not be able to find any additional
independent variables that are significant, except by chance. The Link Test is a regression using the Y estimate and
the Y estimate^2 as the independent variables to regress the dependent variable Y. If your model is properly
specified, the Y estimate independent variable will be statistically significant because it is the predicted value from
the original model. And, the Y estimat^2 will not be statistically significant because if the model is specified
correctly, the squared predictions should not have much explanatory power. And, that is what we got here.
The Y estimate is very statistically significant with a t Stat of 10.1 and a P value of essentially Zero (0.00...).
The Y estimate^2 is not statistically significant with a t Stat of -1.13 and a P value of 0.26.
EconometricsEconometric models.xlsxReg model testing.xlsxModel Specification

Intro to econometrics

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Intro to econometrics