CO-OPERATIVE UNIVERSITY, SAGAING Department of Statistics

APPLIED ECONOMETRICS C Stats 4105

Fourth Year (Second Semester) Reference Johnston, J., & Dinardo, J. Econometric Methods. Fourth Edititon.

CONTENTS  Introduction  Analysis of variance in the two variable Regression model  Time as a Regressor

Introduction What is econometrics? Econometrics uses economic theory, mathematics, and statistical inference to quantify economic phenomena. In other words, it turns theoretical economic models into useful tools for economic policy making.

Analysis of variance in the two –variables regression model (ANOVA) S.V

S.S

Df

MS

F-ratio

X

SSR

K-1

MSR

F=𝑀𝑆𝐸

Residual

SSE

n-k

MSE

Total

SST

n-1

𝑀𝑆𝑅

1. Hypothesis 𝐻0 : X does not play a statistically significant role in the explanation of Y. 𝐻1 ∶ X play a statistically significant role in the explanation of Y. 2. Test statistic 𝑀𝑆𝑅 F = 𝑀𝑆𝐸 3. Critical Value k = 𝐹(𝛼,𝐾−1,𝑛−𝑘) 4. Decision rule If F > 𝐾; reject 𝐻0 Otherwise, accept 𝐻0 5. Decision 6. Conclusion

Test for significance of the slope coefficient 1. Hypothesis 𝐻𝑜 : 𝛽 = 0 𝐻1 : 𝛽 ≠ 0 2. Test statistic 𝑏−0 t = 𝑠.𝑒 (𝑏) 3. Critical value K = 𝑡(𝛼Τ2,𝑛−2) = 𝑡0.025,3 4. Decision rule If 𝑡 ≥ K, reject 𝐻0 Otherwise; accept 𝐻0

5. Decision 𝑡 >K Otherwise reject 𝐻0 6. Conclusion

Test of significance relationship between X and Y 1. Hypothesis 𝐻𝑜 : 𝜌 = 0 (No significantly relationship exist between X and Y) 𝐻1 : 𝜌 ≠ 0 (Significantly relationship exist between X and Y) 2. Test statistic t=

𝑟 𝑛−2 1−𝑟 2

3. Critical value 𝛼 = 0.05 K = 𝑡(𝛼Τ2,𝑛−2) 4. Decision rule If 𝑡 ≥ K, reject 𝐻0 Otherwise; accept 𝐻0 5. Decision 6. Conclusion

Simple linear Regression model Y = 𝛼 + 𝛽𝑋 + 𝑢, 𝑖 = 1,2, … , 𝑛 Y = dependent variable X = independent 𝛼 = intercept term 𝛽 = slope coefficient 𝑢𝑖 = error term 2

ത = ෍ 𝑌𝑖 2 = 86.9, ෍ 𝑌 = 21.9, ෍(𝑌𝑖 − 𝑌) ෍ 𝑋𝑖 − 𝑋ത 𝑌𝑖 − 𝑌ത = ෍ 𝑋𝑌 = 106.4 2

ത = ෍ 𝑋𝑖 2 = 215.4 ෍(𝑋𝑖 − 𝑋) σ𝑥 𝑦 𝛽መ = 𝑏 = σ 𝑖 2𝑖 = 𝑥𝑖

𝛼ෝ = a =

σ 𝑌𝑖 −𝑏 σ 𝑋𝑖 𝑛

106.4 215.4

=

= 0.4939

21.9−(0.4939×186.2) 20

= -3.5032

Fitted trend 𝑌෠ = 𝑎 + 𝑏𝑋𝑖 = -3.5032+0.4939 𝑋𝑖 Standard error of intercept term S.E (a) = 𝑉(𝑎) 1 𝑋ത + σ 𝑋𝑖 2 𝑛 2 2 2 σ 𝑢𝑖 𝜎 = 𝑆 = 𝑛−𝑘 σ 𝑢𝑖 2 = σ 𝑦𝑖 2 − 𝑏 σ 𝑥𝑖 𝑦𝑖

V(a) = 𝜎 2

= 86.9 – (0.4939 × 106.4) = 34.349 34.349 𝑆 2 = 20−2 = 9.31

V(a) = 1.9083

1 (9.31)2 + 215.4 20

= 0.8633

S.E (a) = 0.8633 = 0.9291 Standard error of slope coefficient S.E (b) = 𝑉(𝑏) V(b) =

𝜎2 σ 𝑥𝑖 2

= 0.0089

S.E (b) = 0.0089 = 0.0943

Estimated mean value of Y when X=10, ෠ -3.5032 + (0.4939×10) 𝑌= = 1.4358 95% confidence interval for mean is 1 (𝑋𝑖 − 𝑋)2 𝑌෠ ± 𝑡(∝ൗ ,𝑛−2) 𝑆 + 2 𝑛 σ 𝑋𝑖 2 (1-∝)100% = 95% 1- ∝ = 0.95 ∝= 0.05 ∝ = 0.025 2 S = 𝑆2 = 1.9083 = 1.3814 𝑡(∝Τ2,𝑛−2) = 𝑡

0.025,18 =2.101

LCL = 𝑌෠ − 𝑡(∝Τ2,𝑛−2) 𝑆

1 𝑛

+

(𝑋𝑖 −𝑋)2 σ 𝑋𝑖 2

= 1.4358− 2.101 × 1.3814

= 0.7726

1 10−9.31 2 + 20 215.4

UCL = 𝑌෠ + 𝑡(∝Τ2,𝑛−2) 𝑆

1 𝑛

+

(𝑋𝑖 −𝑋)2 σ 𝑋𝑖 2

= 1.4358+ 2.101 × 1.3814

= 2.099

1 10−9.31 2 + 215.4 20

Time as a regressor 𝑌𝑡 =∝ +𝛽𝑡 + 𝑢𝑡 ,

𝑡 = 1,2, … , 𝑛

t

Odd (t)

Even (t)

1

-3

-5

2

-2

-3

3

-1

-1

.

0

+1

.

1

.

.

2

3

.

3

5

.

.

.

.

.

.

.

.

.

σ𝑡 = 0

σ𝑡 = 0

𝑏=

a=

𝑛 σ 𝑡 𝑌𝑡 − σ 𝑡 σ 𝑌𝑡 2

𝑛 σ 𝑡 2 − (σ 𝑡) σ 𝑌𝑡 −𝑏 σ 𝑡 𝑛

If “t” is odd or even

b= a=

σ 𝑡𝑌𝑡 σ 𝑡2 σ 𝑡𝑌𝑡 𝑛

The model is, 𝑌𝑡 =∝ + β𝑡 + 𝑢𝑡 𝑌𝑡 = The production t = years ∝, 𝛽 = constant 𝑢𝑡 = error term Forecast model ෝ + 𝛽መ 𝑡 𝑌෠𝑡 = ∝ = a + bt

𝒀𝒕

Year

𝒕𝟐

t

t𝒀𝒕

1985

38.1

-2

4

-76.2

1986

80.0

-1

1

-80

1987

170.4

0

0

0

1988

354.5

1

1

354.5

1989

744.4

2

4

1488.8

1387.4

0

0

1687.1

σ 𝑡𝑌 1687.1 𝛽መ = 𝑏 = σ 𝑡 2𝑡 = 10 = 168.71

ෝ=a= ∝

σ 𝑌𝑡 𝑛

=

1387.4 5

= 277.48

𝑌෠𝑡 = 277.48 + 168.71𝑡 For 1995 (t= 8) 𝑌෠𝑡 = 277.48 + 168.71 8 = 1627.16

The estimated Marijuana production for the year 1995 is 1627.16 (10000 tons)