Week 11: Chapters 10 and 11 Flashcards
What are the differences between Time Series and Cross-Sectional data
- Temporal ordering of observations, where the past affects the future
- Time series data represents a random variable (a draw of a stochastic process)
What are the four types of time series models?
- Static models
- Distributed lag models (DL)
- Autoregressive models (AR)
- Autoregressive distributed lah models (ARDL)
What are the two main reasons for using a static model?
- To see if the changes in Xt immediately affect yt
- To see what the trade off is between x and y
What is the B1 in the DL model represent?
The impact propensity (Multiplier)
Write the equation for a DL model
Yt = π½0 + π½1xt + π½2xt-1 π½3xt-2 + ut
Equation for AR model?
Yt = π½0 + π½1yt-1 + ut
What does an AR model do?
Uses past variables of y to explain contemporaneous values of y
What are the first three TS assumptions that result in the OLS estimators being unbiased?
TS1: The regression is linear in its parameters
TS2: No perfect collinearity
TS3: Zero Conditional Mean
Under TS3: What is the formal definition of STRICT exogeneity?
Corr(Xsj, Ut) = 0, even when s does NOT = t
Under TS3: What is the formal definition of Contemporaneous exogeneity?
Corr(Xtj, Ut) = 0, for all values of j
TS1-5 (BLUE)
and TS1-6
- Linear in its Parameters
- No perfect collinearity
- Zero Conditional Mean
- Homoskedasticity
- No serial correlation
- Normality
TS5: No serial correlation definition
Corr (Ut, Us) = 0, for all t does not = s
- Errors are not correlated over time
Why was TS5 not a problem in cross-sectional data?
Due to MLR2, as the random sampling ensured ui and uh were indepdenent
In deterministic trends, what does π½1 >< 0 mean?
π½1 > 0: Upward trend
π½1 < 0: Downward trend
What is a spurious regression?
It is finding a relationship between one or two variables, simply because each is growing over time
Write the final equation for de-trending variables
see notes
Why are R^2 higher in time series?
- Time series data is often in an aggregate form and thus less variance to explain
- Trending dependent variables
When is a stochastic process stationary?
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
Draw the graphs for perfect stationarity, non-constant mean, non-constant variance and seasonality
notes
When does covariance stationarity hold?
- E(xt) is constant
- Var(xt) is constant
- for any t, h, Cov(xt, xt+h) depends only on h and not on t
What is weak dependence?
Places restrictions on how strongly related the random variables xt and xt+h are as h increases
A stationary series is weakly dependent if:
Xt and Xt+h are βalmostβ independent as h increases
- Corr(xt, xt+h) goes to zero as h increases
What does a weakly dependent time series look like?
- Xt:
xt = et + Ξ±(et-1)
xt: Moving average process of order one (MA)
- weighted average of et and et-1
What does a weakly dependent time series look like?
- Yt:
Yt: p1yt-1 + et
- The autoretrogressive process of order one, AR(1)
- When |p1| < 1, the AR(1) process is stable and weakly dependent
Can a trending series be weakly dependent?
Yes, as long as the time trend is controlled for
What is a trend stationary process?
A series that is stationary around its time trend and weakly dependent
New TS assumptions under asymptotic properties of OLS:
TS1β: Linearity and weak dependence
TS2: No perfect collinearity
TS3β: Zero Conditional Mean
TS4β: Homoskedasticity
TS5β: No serial correlation
What happens when TS1β - TS3β hold?
OLS estimators are consistent and PlimE(b^j) = bj
TS3β
Zero conditional mean,
Now its contemporaneous exogeneity such that E(ut, xt) = 0
TS4β
homoskedasticity,
Var(ut, xt) = Ο^2
Errors are contemporaneously homoskedastic
TS5β
No serial correlation
Corr(ut, us|xt, xs) = 0
What happens when TS1β-5β HOLD?
OLS estimators are asymptotically normally distributed
Whats the most common violation of weak dependence?
Problems will arise from highly persistent data
Random walk equation
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
Show that the variance of yt changes over time in a random walk
notes
What are the two main implications of the random walk?
- Random walks is not covariance stationary
- Random walks are not weakly dependent
Random walk with a drift
yt = Ξ±0 + yt-1 + et, where Ξ±0 is the drift term
What does an I0 weakly dependent series mean?
- 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
Whats the benefit of differencing a time series?
- It removes any time trends
- See notes for more
