HANDOUT 1 Flashcards
How do we get a de-trended GDP series? What does it show?
Plot the residuals - what’s left in GDP having extracted the time trend. Shows business cycles.
seasonality =
fluctuations in the data with a regular period
If we extract the time trend and seasonal trend from a series, what are we left with?
Economic series - cyclical elements = explained by economics e.g. income, prices. We’ve extracted the deterministic elements.
Adjustment to equilibrium if NO lags
If X1 increases by 1 unit, Y adjusts immediately to the new equilibrium.
If we only have lags on X variable, how does a change in income today affect the model?
In each period, there’s a direct effect of the change in income on consumption in that period (no indirect). smooth consumption over time. Eventually the shock dissipates out the system
2 methods of finding LR effect
- sum up SR marginal responses
2. solve for LR equilibrium equation
A lagged dependent variable picks up…
Persistence of habits.
Today’s consumption choice affects tomorrow’s consumption choice. Suppose we find £1 today and buy a Mars bar, affects tomorrow’s consumption as now addicted to mars bars.
What is dYt-1 / dX1t if the shock is unexpected?
If the shock is unexpected, a change in income today has NO effect on yesterday’s consumption.
Sum of geometric series =
a / (1 - r)
The LR effect of an increase in income today tells us
The total additional increase in consumption over an individual’s lifetime from finding £1 today over what it would’ve been without the income shock.
What do we implicitly assume about the coefficient on the lagged dependent variable? Why?
We assume the coefficient on the lagged dependent variable is < 1 in absolute magnitude. Condition for stability. We need the indirect effects to get smaller over time so that the shock eventually dissipates out of the system.
Therefore, the persistence of shocks over time is determined by…
The coefficient on the lagged dependent variable. Large coefficient = more persistent shocks.
If the coefficient on the lagged dependent variable is positive (but < 1), what do the dynamics of returning to equilibrium look like?
Smooth decay back to 0 at rate 0.5. Shock eventually dissipates out of the system.
If the coefficient on the lagged dependent variable is negative (but > -1), what do the dynamics of returning to equilibrium look like?
Oscillations between +VE and -VE - but the shock will eventually dissipate out of the system as long as the coefficient on the lagged dependent variable is not more negative that -1.
Therefore, the adjustment back to equilibrium is determined by (2)…
- The sign of the lagged dependent variable
2. The magnitude “
How do we estimate models with only lagged X variables?
STANDARD OLS
Which CLRM assumption is violated when we have a lagged dependent variable?
E(€t I explanatory variables) = 0 Since COV(Yt, €t) ≠ 0 And COV(Yt, Yt-1) ≠ 0 So COV(Yt-1, €t) ≠ 0 - Yt-1 is an explanatory variable here = CLRM assumption 1 violated.
When we have a lagged dependent variable, what do we replace CLRM assumption 1 with?
CLRM assumption 1 = strict exogeneity Replace with Contemporaneous exogeneity E(€t I Y1, Y2,...,Yt-1) = 0 Error term is uncorrelated with PAST values of Y. But it can be correlated with Yt. Still assume E(€t I Xts) = 0
Is OLS unbiased for lagged dependent variables?
NO - but it IS CONSISTENT
As t–> infinity, E(b1)–>B1
When we have a lagged dependent variable model, how do we test single and multiple restrictions?
Single = approx Z test via CLT Multiple = Chi-squared test
What assumption that we had for cross-sectional data do we get rid of? Why?
Get rid of assumption of random sampling, because with time-series data observations are related across time.
What 2 conditions do we replace the random sampling assumption with?
- Stationarity
2. Weak dependence
3 conditions for stationarity
- E(Yt) = M - mean constant over time
- V(Yt) = sigma^2 Y - variance constant over time
- COV(Yt, Yt-h) = gamma h - covariance of the variable between 2 points in time depends only on how far apart, not location in space.
Weak dependence condition
COV(Yt, Yt-h) –> 0 as h–> infinity
A variable today is NOT really driven by what it was at some infinite point in the past.
We can do some time-series analysis only if…
We are prepared to assume stationarity & weak dependence (even though they’re very unrealistic for most series)
