Econometrics – lecture 4 – statistical inference

In practice you have to estimate it.TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 1.0 1.1 0.9 0.8 0.7 as given... TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT Su

Dr TU Thuy Anh Faculty of International Economics

1

KTEE 310 FINANCIAL ECONOMETRICS

STATISTICAL INFERENCE Chap 5 & 7 – S & W

0 :   

H

0 2 2

1 :   

H

We will suppose that we have the standard simple regression model and that

we wish to test the hypothesis H0 that the slope coefficient is equal to some value 20 We test it against the alternative hypothesis H1, which is simply that 2 is not equal to 20

We will test the hypothesis that the rate of price inflation is equal to the rate

of wage inflation The null hypothesis is therefore H0: 2 = 1.0.

0 2 2

0 :   

H

0 1 : 2

0  

H

0 1 : 2

1  

H

0 2 2

1 :   

H

We will assume that we know the standard deviation and that it is equal to 0.1 This is a very unrealistic assumption In practice you have to estimate it.

TESTING A HYPOTHESIS RELATING TO A REGRESSION

COEFFICIENT

1.0 1.1 0.9

0.8 0.7

as given)

TESTING A HYPOTHESIS RELATING TO A REGRESSION

COEFFICIENT

Suppose that we have a sample of data for the price inflation/wage inflation

model and the estimate of the slope coefficient, b2, is 0.9 Would this be

evidence against the null hypothesis 2 = 1.0?

And what if b2 =1.4?

1.0 1.1 0.9

0.8 0.7

as given)

TESTING A HYPOTHESIS RELATING TO A REGRESSION

TESTING A HYPOTHESIS RELATING TO A REGRESSION

COEFFICIENT

For example, we might choose to reject the null hypothesis if it implies that

the probability of getting such an extreme estimate is less than 0.05 (5%).

According to this decision rule, we would reject the null hypothesis if the estimate fell in the upper or lower 2.5% tails.

TESTING A HYPOTHESIS RELATING TO A REGRESSION

COEFFICIENT

The 2.5% tails of a normal distribution always begin 1.96 standard

deviations from its mean Thus we would reject H0 if the estimate were 1.96 standard deviations (or more) above or below the hypothetical mean.

Or if the difference, expressed in terms of standard deviations, were more than 1.96 in absolute terms (positive or negative).

0 :  

H

s.d

96.1

0 2

2   

96.1s.d

/)(b2  20  (b2  20)/s.d.  1.96

TESTING A HYPOTHESIS RELATING TO A REGRESSION

2  

 b z

s.d

96.1

0 2

2   

0 2 2

0 :  

H

The range of values of b2 that do not lead to the rejection of the null

hypothesis is known as the acceptance region.

Type II error (the probability of accepting the false hypothesis)

The limiting values of z for the acceptance region are 1.96 and -1.96 (for a

5% significance test).

acceptance region for b2:

s.d

96.1s.d

96

TESTING A HYPOTHESIS RELATING TO A REGRESSION

acceptance region for b 2

probability of making a Type I error if the null hypothesis is true.

Type I error: rejection of H0 when it is in fact true Probability of Type I error: in this case, 5%

Significance level of the test is 5%.

TESTING A HYPOTHESIS RELATING TO A REGRESSION

acceptance region for b 2

t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT

We replace the standard deviation in its denominator with the standard

error, the test statistic has a t distribution instead of a normal distribution

We look up the critical value of t and if the t statistic is greater than it,

positive or negative, we reject the null hypothesis If it is not, we do not.

s.e

0 2

A graph of a t distribution with 10 degrees of freedom When the number of degrees of freedom is large, the t distribution looks very much like a normal

distribution

00.10.20.30.4

t distribution has longer tails than the normal distribution, the difference

being the greater, the smaller the number of degrees of freedom

This means that the rejection regions have to start more standard deviations

away from zero for a t distribution than for a normal distribution.

normal

00.1

The 2.5% tail of a t distribution with 10 degrees of freedom starts 2.33

standard deviations from its mean.

That for a t distribution with 5 degrees of freedom starts 2.57 standard

deviations from its mean.

normal

00.1

t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT

For this reason we need to refer to a table of critical values of t when

performing significance tests on the coefficients of a regression equation.



t Distribution: Critical values of t

Degrees of Two-sided test 10% 5% 2% 1% 0.2% 0.1% freedom One-sided test 5% 2.5% 1% 0.5% 0.1% 0.05%

Number of degrees of freedom in a regression

= number of observations – number of parameters estimated.

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82 0 21

1

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1 10

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( s.e. 2

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of degrees

2

% 5 , crit 

t

1 :

; 1

H

u w

Log-likelihood -89,96960 Akaike criterion 185,9392Schwarz criterion 189,4734 Hannan-Quinn 186,8768rho 0,098491 Durbin-Watson 1,535082

EXAMPLE

Log-likelihood -89,58910 Akaike criterion 187,1782Schwarz criterion 191,8904 Hannan-Quinn 188,4284rho -0,083426 Durbin-Watson 1,737618

EXAMPLE

Hypothesis testing using p-value

 Step 1: Calculate tob =

 Step 2: Calculate p-value = P (|t| > |tob|)

 Step 3: Gor a given α:

• Two-tail test: p-value < α  reject H0

• One-tail test: p-value/2 < α:  reject H0

20

) ˆ (

ˆ

i

i i

se 



 

Log-likelihood -89,96960 Akaike criterion 185,9392Schwarz criterion 189,4734 Hannan-Quinn 186,8768rho 0,098491 Durbin-Watson 1,535082

EXAMPLE

Log-likelihood -89,58910 Akaike criterion 187,1782Schwarz criterion 191,8904 Hannan-Quinn 188,4284rho -0,083426 Durbin-Watson 1,737618

EXAMPLE

probability density function of b2

(1) conditional on 2 = 2 being true

(2) conditional on 2 = 2 being true

min

min max

The diagram shows the limiting values of the hypothetical values of 2, together with their associated probability distributions for b2.

2 2 + 1.96sd

max2 - sd maxmax2+sd max

(1)(2)

Any hypothesis lying in the interval from 2min to 2max would be compatible with the sample estimate (not be rejected by it) We call this interval the 95% confidence interval.

reject any 2 > 2 = b2 + 1.96 sdreject any 2 < 2 = b2 - 1.96 sd

95% confidence interval:

b2 - 1.96 sd < 2 < b2 + 1.96 sd

max min

CONFIDENCE INTERVALS

Standard deviation known95% confidence interval

b2 - 1.96 sd < 2 < b2 + 1.96 sd99% confidence interval

b2 - 2.58 sd < 2 < b2 + 2.58 sdStandard deviation estimated by standard error95% confidence interval

b2 - tcrit (5%) se < 2 < b2 + tcrit (5%) se99% confidence interval

b2 - tcrit (1%) se < 2 < b2 + tcrit (1%) se

* ob

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b se

b

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EXAMPLE

SIGNIFICANCE OF MODEL – F TEST

 If the model is significant?

 H0: R2 =0; H1: R2>0

 If Fob > Fα(k-1,n-k)  reject H0

2 2

/ ( 1) (1 ) / ( )

F(1, 22) 204,2536 P-value(F) 1,29e-12

Log-likelihood -96,26199 Akaike criterion 196,5240Schwarz criterion 198,8801 Hannan-Quinn 197,1490rho 0,836471 Durbin-Watson 0,763565

EXAMPLE

F(2, 21) 171,9278 P-value(F) 9,57e-14

Log-likelihood -89,96960 Akaike criterion 185,9392Schwarz criterion 189,4734 Hannan-Quinn 186,8768rho 0,098491 Durbin-Watson 1,535082

EXAMPLE

F-TEST

 Y = 1 + 2X2+ + 5X5 + u (1)

 if 2 = 4=0 ?

H0: 2 = 4=0; H1: at least one of them is nonzero

Step1: run unrestricted model (1) => R2(1)

Step 2: run: restricted model: Y = 1 + 3X3+ 5X5 + u (2) => R2(2)