L7: TS continued Flashcards
What is the problem with using sequential t or F-tests to select lag length? (2)
They tend to choose models that are too large and the process is cumbersome
2 information criteria that can be used to select lag length?
1) Bayesian IC (BIC)
2) Akaike IC (AIC)
Explain how we use the Bayesian IC?
We minimise BIC(p), because this trades off bias and variance to determine a ‘best’ value of p for the forecast (see notes!!!)
Main difference between BIC and AIC?
AIC has a lower penalty term (the second term) for using more parameters since 2/T is less than ln(T)/T, therefore the AIC will estimate a greater number of lags are needed (larger p)
When can an AIC be more useful as opposed to a BIC?
If we believe LT lags are important!
2 problems with the AIC estimator?
1) It isn’t consistent (BIC is consistent)
2) It can overestimate p
What is the problem with choosing R-squared to choose the number of lags?
We would always choose the largest possible number of lags (see slide 6)
See
‘in practice’ halfway down P1S1
What happens if the assumption of stationarity doesn’t hold?
NONstationary series
2 types of nonstationarity?
1) Trends
2) Structural breaks
What is a trend?
A persistent, LT movement/tendency in the data (not necessarily a straight line though)
See
Slides 10 and 11 show a trend, slide 12 does not, slide 13 explains as follows:
Log Japan GDP clearly has a long-run trend – not a straight line, but a slowly decreasing trend – fast growth during the 1960s and 1970s, slower during the 1980s, stagnating during the 1990s/2000s.
• Inflation has long-term swings, periods in which it is persistently high for many years (’70s/early ’80s) and periods in which it is persistently low. Maybe it has a trend – hard to tell.
• NYSE daily changes has no apparent trend. There are periods of persistently high volatility – but this isn’t a trend.
What is a deterministic trend? (uncommon in economics)
A non-random function of time (eg. yt=t or yt=t^2)
What is a stochastic trend? Give an example.
A stochastic trend is random and varies over time (eg. a random walk)
How, mathematically, can we describe a random walk? (3 points, important to know this well!)
A random walk is a AR(1) model where beta1=1 because there is equal chance the series will tend up in the next period as there is it will tend down (tf is not included in the model)
Yt=Y(t-1)+ut where ut is SERIALLY UNcorrelated
ie. if Yt follows a random walk, the value tomorrow equals that of today plus an unpredictable disturbance!
2 key features of a random walk, explained?
1) Y(T+h|T) = Y(T)
This states that the best prediction of Y tomorrow/next week/next month is the value today!
2) Suppose Y0=0, then var(Yt)=tσ^2
This means that the variance depends on t tf increases linearly with t tf Yt isn’t stationary!!!
Is ut serially uncorrelated for a RW with a drift?
Yes it is!
If for a random walk with drift, B0 is not equal to zero, what does this mean?
Yt follows a RW around a linear trend!
Note
A RW with trend is a good description of many stochastic trends in economic TS, but not a good predictor since is random
Are random walks stationary, or nonstationary? What problem does this create, and how can we solve it?
A RW is nonstationary
This creates a problem that it is not a good predictive model since it is random
SOLUTION:
If Yt has a RW trend, then ΔYt IS stationary and regression analysis should be undertaken using ΔYt instead of Yt
3 main problems caused by trends?
1) AR coefficients are strongly biased towards zero tf -> poor forecasts
2) Some t-statistics don’t have a standard normal distribution (even with n large)
3) Spurious regression
What is spurious regression?
If X and Y both have RW trends then they may look related even though they are not (slide 19 and 20, panopto 6th december 4mins in)
2 steps to detecting stochastic trends?
1) Plot the data - are there persistent LR movements?
2) Use a regression-based test for a RW: the Dickey-Fuller test for a UNIT ROOT
Given the autoregressive process:
Yt=β0+β1Y(t-1)+ut
How does a DF-test if this a RW or not?
If β1=1 then we have a RW with drift tf stochastic trend with a unit root tf cannot do empirical analysis on this! TF:
ΔYt=β0+𝛿1Y(t-1)+ut -> we can estimate this!
Hypotheses:
H0: 𝛿1=0 H1: 𝛿1<0
IF 𝛿1<0 then Yt is stationary tf use AR(1) model, if 𝛿1=0 then β1=1 tf RW tf stochastic trend and unit root!
How do we actually compute a DF test?
Compute the t-stat. testing if 𝛿1=0 - BUT under H0, this t-statistic is not normally distributed! Tf need to use DF critical values! There are 2 cases, which have different CVs:
a) ΔYt=β0+𝛿1Y(t-1)+ut (INTERCEPT ONLY)
b) ΔYt=β0+μt+𝛿1Y(t-1)+ut (INTERCEPT AND TIME TREND)
What is μt in the intercept and time trend specification of the DF test?
μ is a parameter
t is time index t=1,…,T (deterministic variable)
What do we need to do a DF test?
Large sample
Explain what the Augmented DF test is?
DF test is only valid if u(t) is a white noise; however in TS data there is often autocorrelation therefore the errors are unlikely to be pure white noises
Solution: augment the test using p lags of the dependent variable (see notes?)
When do you use the intercept and time trend vs. just intercept specifications of the DF test?
Intercept and TT: LT growth in trend
Intercept only: no LT growth in trend
Solution to a unit root?
If Yt has a unit root (stochastic RW trend) the easiest way around this problem is to model Yt in first differences
Explain summary of detecting and addressing stochastic trends?
1) Plot Yt; if it looks like it has a trend/trend is plausible, compute DF test (either one)
2) If DF accepts H0, conclude non-stationarity (unit root), if it rejects H0, conclude stationarity
3) If Yt has a unit root, use change in Yt for regression analysis and forecasting; if no unit root, use Yt
What are structural breaks?
Type of nonstationarity; means the coefficients of the model might be different at different points in time!
Issue of structural breaks?
Can cause model to not be externally valid
2 cases of testing of structural breaks?
1) Break date is known
2) Break date is unknown (finish)
