Submitted to: Dr. Tanweer ul Islam
Submitted by: Kaneez Fatima
MS ECONOMICS 2K20
NUST SCHOOL OF SOCIAL SCIENCES & HUMANITIES
Applied Econometrics Assignment # 2
Q. Proof
𝒄𝒐𝒓𝒓(𝒚𝒕 , 𝒚𝒕−𝟐 ) = 𝒂𝟐𝟏
𝒄𝒐𝒓𝒓(𝒚𝒕 , 𝒚𝒕−𝟏 ) = 𝒂𝟏
Solution:
Before giving the proof, we have to check the data for stationarity
If it is stationary, then the mean value will remain constant over time i.e. 𝜇 = 0. If 𝒂𝟏 is constant,
then its mean value is also constant; the absolute value of 𝒂𝟏 is less than 1.
Using autoregressive model AR (1):
Let 𝒚𝒕 =𝒂𝟏 𝒚𝒕−𝟏 + 𝜺𝒕 i.e., put 𝒂𝟏 = 𝟎 → 𝝁 = 𝟎
𝒄𝒐𝒗(𝒚𝒕 , 𝒚𝒕−𝟏 ) = 𝚬[(𝒂𝟏 𝒚𝒕−𝟏 + 𝜺𝒕 )𝒚𝒕−𝟏 ]
𝒄𝒐𝒗(𝒚𝒕 , 𝒚𝒕−𝟏 ) = 𝒂𝟏 𝚬(𝒚𝟐𝒕−𝟏 )
𝒄𝒐𝒗(𝒚𝒕 , 𝒚𝒕−𝟏 ) = 𝒂𝟏 𝝈𝟐𝒚
𝒄𝒐𝒓𝒓(𝒚𝒕 , 𝒚𝒕−𝟏 ) =
𝒄𝒐𝒗(𝒚𝒕 , 𝒚𝒕−𝟏 )
𝒗𝒂𝒓(𝒚𝒕 , 𝒚𝒕−𝟏 )
Due to stationarity, the variance of 𝒚𝒕 = the variance of 𝒚𝒕−𝟏
𝒄𝒐𝒓𝒓(𝒚𝒕 , 𝒚𝒕−𝟏 ) = 𝒂𝟏
𝒄𝒐𝒗(𝒚𝒕 , 𝒚𝒕−𝟐 ) = 𝚬[(𝒂𝟏 𝒚𝒕−𝟏 + 𝜺𝒕 )𝒚𝒕−𝟐 ]
𝒄𝒐𝒗(𝒚𝒕 , 𝒚𝒕−𝟐 ) = 𝚬[(𝒂𝟏 (𝒂𝟏 𝒚𝒕−𝟐 + 𝜺𝒕−𝟏 ) + 𝜺𝒕 )𝒚𝒕−𝟐 ]
𝒄𝒐𝒗(𝒚𝒕 , 𝒚𝒕−𝟐 ) = 𝚬(𝒂𝟐𝟏 𝒚𝟐𝒕−𝟐 + 𝒂𝟏 𝜺𝒕−𝟏 𝒚𝒕−𝟐 + 𝜺𝒕 𝒚𝒕−𝟐 )
𝒄𝒐𝒗(𝒚𝒕 , 𝒚𝒕−𝟐 ) = 𝒂𝟐𝟏 𝝈𝟐𝒚
Due to stationarity, the variance of 𝒚𝒕 = the variance of 𝒚𝒕−𝟐
𝒄𝒐𝒓𝒓(𝒚𝒕 , 𝒚𝒕−𝟐 ) =
𝒄𝒐𝒗(𝒚𝒕 , 𝒚𝒕−𝟐 )
𝒗𝒂𝒓(𝒚𝒕 , 𝒚𝒕−𝟐 )
𝒄𝒐𝒓𝒓(𝒚𝒕 , 𝒚𝒕−𝟐 ) = 𝒂𝟐𝟏