econometric modelling. introduction to examine some econometric results from various financial...
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![Page 1: Econometric Modelling. Introduction To examine some econometric results from various financial models To use the results to determine levels of significance](https://reader035.vdocument.in/reader035/viewer/2022070306/5515df0b55034638038b4b2a/html5/thumbnails/1.jpg)
Econometric Modelling
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Introduction
• To examine some econometric results from various financial models
• To use the results to determine levels of significance of the variables and whether the results fit the theory
• To use the results for testing specific restrictions.
• Suggest some potential problems when assessing model results
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Carrying out a regression
• Set out the model/theory, including expected signs and magnitudes of the coefficients
• Gather data• Estimate the model using a relevant
technique• Interpret results, assess diagnostic tests.• If model fails the diagnostic tests,
respecify model
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Stock price return Model• Given the following model, We wish to obtain
estimates of the constant and slope coefficients:
0,0,00,0
: thatassume wewhere
4321
43210
or
urpipys tttttt
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Variables
premiumrisk
rateinterest
inflation
production industrial
parameter slope-
constant
pricesstock
t
t
t
t
t
rp
i
p
y
s
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Estimation
• We would estimate this model using ordinary least squares (OLS), although as we will find out later other methods may be more appropriate.
• The model is estimated using monthly data from 2000m1 to 2005 m12.
• This gives 6 years of data producing 72 observations.
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Results
s)parenthesein errors standard(
2.15)14( '
8.12)12(
25)67,4(,84.1,58.0R
(0.3) (0.01) (0.2) (0.2) (0.2)
7.01.02.08.04.0ˆ
2
testsWhite
LM
FDW
rpipys ttttt
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Coefficients
• The signs on the coefficients are as we hypothesised with the possible exception of p.
• However as this variable is insignificant, the sign is of less importance.
• For y, a unit rise in y gives a 0.8 of a unit rise in the dependent variable s(t). For p, a unit rise in p gives a 0.2 of a unit rise in s(t) etc.
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T-statistics
• Firstly test if the 4 variables are individually different to 0, using the t-test (we usually ignore the constant)
• E.g. y: 0.8-0/0.2=4• Critical value is 2.000 (5%) (72-5 degrees of freedom, 60
d of f is nearest in tables)• As 4>2 we reject the null hypothesis that y=0, therefore y
is said to be significantly different to 0.• The t-statistics for p: 1, i: 10 and rp:2.333, we conclude
that y, i and rp are significant and p is insignificant.• This result would suggest we might consider removing p
from our model.
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R-squared
• The adjusted R-squared statistic is 0.58, which is relatively good explanatory power.
• The F-test for the significance of the goodness of fit is 25. The critical value for F(4,67) is 2.53 (5%).
• As 25 > 2.53, the goodness of fit of the regression is significant, or the joint explanatory power of the variables is significantly different to 0.
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DW statistic
• We first need to find the dl and du values from the tables. As k is 4 and we have 72 observations, the critical values are: dl-1.49 and du 1.74.
• The DW statistic is 1.84, which is between du (1.74) and 4-du (2.26), so we accept the null hypothesis of no 1st order autocorrelation.
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LM test for higher autocorrelation
• Given that we have monthly data, we test for 12th order autocorrelation.
• The critical value for chi-squared (12) is 21.026.
• As the LM statistic is 12.8 < 21.026, we accept the null hypothesis of no 12th order autocorrelation
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White’s test for heteroskedasticity
• This follows a chi-squared distribution with 14 degrees of freedom (including cross product terms)
• The critical value with 14 degrees of freedom is: 23.685 using the chi-squared tables
• As 15.2 < 23.685, we accept the null hypothesis that there is no heteroskedasticity.
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Market Model• According to the market model, the return
on an asset is determined by a constant and the return to the market index.
error terman is
indexmarket a return to theis R
parameter slope a is
constant a is
iasset an return to theis
mt
t
it
tmtit
u
R
uRR
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Market Model
• As before we would wish to run an OLS regression, then interpret the coefficient, t-statistics and various diagnostic tests for autocorrelation and heteroskedasticity.
• In this model we would expect β > 0, the closer to 1, the closer asset i follows the market index.
• If we have 100 days of daily data for the regression, we get the following result:
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Market Model
s)parenthesein errors (standard
(0.1) (0.4)
9.06.0ˆmtit RR
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Market Model
• The result shows that a unit rise in the market produces a 0.9 of a unit rise in asset i. This suggests this asset closely follows the market and would be considered safe.
• The t-statistic shows that the market index is significant, 0.9-0/0.1=9. critical value is 1.98. As 9 > 1.98, we reject the null hypothesis and say that the market has a significant effect on asset i.
• The usual diagnostic tests would have been produced and interpreted as before.
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The Market-Adjusted-Returns-Model
• Based on the original model, termed the market model, we can test a restriction using the t-test to determine if the model we have is an alternative specification, termed the market-adjusted-returns model.
mtit RR
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Test
• This model implies the following:
β=1
• We can use a t-test to determine if this condition holds.
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Test
• A t-test can also be used to determine if β = 1.
• The critical value is the same as before, the test is:
11.0
19.0
)ˆ(
1ˆ
se
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Test
• As 1< 2 (ignore the sign the t-statistic is an absolute value), we fail to reject the null hypothesis and conclude that the market adjusted model applies.
• In this case the hypothesis is:
1:
1:
1
H
Ho
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F-test of a restriction
• The main use of this F-test is to determine if a group of explanatory variables are jointly equal to 0.
• However we can test alternative theories or restrictions
• The most common restriction is that 2 or more explanatory variables sum to 1.
• The best example of this is the Cobb-Douglas production function.
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Cobb-Douglas Production Function
• This model suggests that output is a function of capital and labour, in logarithmic form it can be expressed as:
error terman is
labour is
capital is
output is
0,0:
logloglog
21
210
t
t
t
t
tttt
u
l
k
y
where
ulky
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Restriction
• We can then test if constant returns to scale applies by testing the restriction that the coefficients on the capital and labour coefficients sum to 1.
• Constant returns to scale are a proportionate increase in all inputs produces a proportionate increase in outputs.
• This allows us to rewrite the Cobb-Douglas production function in terms of output per unit of labour (divide through by l)
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Test for Constant Returns to Scale
• Run the regression in its unrestricted form with both explanatory variables (k and l)
• Collect the RSS (unrestricted)• Run the following restricted version, with constant
returns to scale and again collect the RSS (restricted), then use the formula used previously (see over)
tt
tt
t ulk
ly )log()log( 10
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Test for constant returns to scale Cont..
ondistributi
m
knRSS
mRSSRSSF
u
UR
mk-nF
) (1 nsrestrictio ofnumber
/
/
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Test for constant returns to scale….
• If we get a RSS (unrestricted) of 1.2 and a RSS (restricted) of 1.4 and we have 60 observations. We would get an F statistic of:
1:
1:
:hypothesis nullreject so 45.9
4
5.9360/2.1
1/2.14.1
211
210
157
H
H
F
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Constant returns to scale…
• We would reject the null hypothesis of constant returns to scale, therefore we would use the unrestricted model with capital and labour included separately. The null hypothesis is:
0:
0:
111
210
H
H
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Conclusion
• When running an OLS regression, we need to assess the coefficients, t-statistics and diagnostic tests.
• We can also use the t-statistic to determine if a coefficient equals 1.
• The F-test can also be used to test a specific restriction in a model.