Week 11: Chapters 10 and 11

Question 1

What are the differences between Time Series and Cross-Sectional data

Answer 1

• Temporal ordering of observations, where the past affects the future
• Time series data represents a random variable (a draw of a stochastic process)

Question 2

What are the four types of time series models?

Answer 2

• Static models
• Distributed lag models (DL)
• Autoregressive models (AR)
• Autoregressive distributed lah models (ARDL)

Question 3

What are the two main reasons for using a static model?

Answer 3

• To see if the changes in Xt immediately affect yt
• To see what the trade off is between x and y

Question 4

What is the B1 in the DL model represent?

Answer 4

The impact propensity (Multiplier)

Question 5

Write the equation for a DL model

Answer 5

Yt = 𝛽0 + 𝛽1xt + 𝛽2xt-1 𝛽3xt-2 + ut

Question 6

Equation for AR model?

Answer 6

Yt = 𝛽0 + 𝛽1yt-1 + ut

Question 7

What does an AR model do?

Answer 7

Uses past variables of y to explain contemporaneous values of y

Question 8

What are the first three TS assumptions that result in the OLS estimators being unbiased?

Answer 8

TS1: The regression is linear in its parameters
TS2: No perfect collinearity
TS3: Zero Conditional Mean

Question 9

Under TS3: What is the formal definition of STRICT exogeneity?

Answer 9

Corr(Xsj, Ut) = 0, even when s does NOT = t

Question 10

Under TS3: What is the formal definition of Contemporaneous exogeneity?

Answer 10

Corr(Xtj, Ut) = 0, for all values of j

Question 11

TS1-5 (BLUE)
and TS1-6

Answer 11

1. Linear in its Parameters
2. No perfect collinearity
3. Zero Conditional Mean
4. Homoskedasticity
5. No serial correlation
6. Normality

Question 12

TS5: No serial correlation definition

Answer 12

Corr (Ut, Us) = 0, for all t does not = s
- Errors are not correlated over time

Question 13

Why was TS5 not a problem in cross-sectional data?

Answer 13

Due to MLR2, as the random sampling ensured ui and uh were indepdenent

Question 14

In deterministic trends, what does 𝛽1 >< 0 mean?

Answer 14

𝛽1 > 0: Upward trend
𝛽1 < 0: Downward trend

Question 15

What is a spurious regression?

Answer 15

It is finding a relationship between one or two variables, simply because each is growing over time

Question 16

Write the final equation for de-trending variables

Answer 16

see notes

Question 17

Why are R^2 higher in time series?

Answer 17

1. Time series data is often in an aggregate form and thus less variance to explain
2. Trending dependent variables

Question 18

When is a stochastic process stationary?

Answer 18

If the probability distribution of the stochastic process does not change over time
- If we take any stochastic process and shift it ahead h periods of time, the joint probability distribution looks the same
- The correlation between adjacent terms is the same across all time periods
- Constant mean, constant variance and no trends/seasonality

Question 19

Draw the graphs for perfect stationarity, non-constant mean, non-constant variance and seasonality

Answer 19

notes

Question 20

When does covariance stationarity hold?

Answer 20

• E(xt) is constant
• Var(xt) is constant
• for any t, h, Cov(xt, xt+h) depends only on h and not on t

Question 21

What is weak dependence?

Answer 21

Places restrictions on how strongly related the random variables xt and xt+h are as h increases

Question 22

A stationary series is weakly dependent if:

Answer 22

Xt and Xt+h are ‘almost’ independent as h increases
- Corr(xt, xt+h) goes to zero as h increases

Question 23

What does a weakly dependent time series look like?
- Xt:

Answer 23

xt = et + α(et-1)
xt: Moving average process of order one (MA)
- weighted average of et and et-1

Question 24

What does a weakly dependent time series look like?
- Yt:

Answer 24

Yt: p1yt-1 + et
- The autoretrogressive process of order one, AR(1)
- When |p1| < 1, the AR(1) process is stable and weakly dependent

Question 25

Can a trending series be weakly dependent?

Answer 25

Yes, as long as the time trend is controlled for

Question 26

What is a trend stationary process?

Answer 26

A series that is stationary around its time trend and weakly dependent

Question 27

New TS assumptions under asymptotic properties of OLS:

Answer 27

TS1': Linearity and weak dependence TS2: No perfect collinearity TS3': Zero Conditional Mean TS4': Homoskedasticity TS5': No serial correlation

Question 28

What happens when TS1' - TS3' hold?

Answer 28

OLS estimators are consistent and PlimE(b^j) = bj

Question 29

TS3'

Answer 29

Zero conditional mean, Now its contemporaneous exogeneity such that E(ut, xt) = 0

Question 30

TS4'

Answer 30

homoskedasticity, Var(ut, xt) = σ^2 Errors are contemporaneously homoskedastic

Question 31

TS5'

Answer 31

No serial correlation Corr(ut, us|xt, xs) = 0

Question 32

What happens when TS1'-5' HOLD?

Answer 32

OLS estimators are asymptotically normally distributed

Question 33

Whats the most common violation of weak dependence?

Answer 33

Problems will arise from highly persistent data

Question 34

Random walk equation

Answer 34

yt = yt-1 + et, y0 is independent of et for t>1 - yt is an accumulation of all past shocks and an initial value - the effect of the shocks will be contained in the series forever

Question 35

Show that the variance of yt changes over time in a random walk

Answer 35

notes

Question 36

What are the two main implications of the random walk?

Answer 36

- Random walks is not covariance stationary - Random walks are not weakly dependent

Question 37

Random walk with a drift

Answer 37

yt = α0 + yt-1 + et, where α0 is the drift term

Question 38

What does an I0 weakly dependent series mean?

Answer 38

- I0: Integrated of order zero - Nothing needs to be doe to the series to use it in the regression - Their averages satisfy the standard limit theorems

Question 39

Whats the benefit of differencing a time series?

Answer 39

- It removes any time trends - See notes for more