Transcript Advanced Time Series
Advanced Time Series
PS 791C
Advanced Time Series Techniques
• A number of topics come under the general heading of “state-of-the-art” time series – Unit Root tests – Granger Causality – Vector Autoregression Models – Error Correction Models – Co-Integration Models – Fractional Integration
Nested Special Cases
• Many of these techniques can be considered a more general version of others.
• For instance – OLS is a special case of ARIMA – An ARIMA Model is a Special Case of an SEQ model – An SEQ model is a special case of a VAR
Trend Stationary Processes
• A Simple Linear trend
y t
t
u t
• This can be differenced to eliminate the trend
y t
y t
y t
1
u t
u t
1 • Differencing once more removes the β and therefore make the series stationary 2
y t
2
u t
u t
2
u t
1
u t
2
Difference Stationary Processes
• Suppose that we have a slightly different process
y t
y t
1
t
• Also known as a random walk
Implications
• If we estimate the wrong model there are severe consequences for regression – Regression of a random walk on time will produce an R 2 of about .44 regardless of sample size, even when there is actually no relationship at all – T-tests are not valid – The residuals are autocorrelated – Subject to spurious regression
Unit Root Tests
• In order to avoid this, we need to know if the series is a DSP or TSP process • This means that we are testing whether =1.0, and hence has become known as a Unit Root test – The Dickey-Fuller test – The Augmented Dickey-Fuller Test – The Phillips-Perron test
Dickey-Fuller test
• The Dickey-Fuller test requires estimating the following model
y t
y t
1
t
t
• The series is a DSP if =1 and β=0, and a TSP if | |<1 • Cannot use least squares, so they employ a LR test, and provide tables
CoIntegration
• A model in which the X and Y variables have unit root processes is called a cointegrated process.
• Such models are exceedingly likely to exhibit spurious correlation and will likely have non-stationary residuals.
Granger Causality
• Ordinary regression tests correlation • Causation is implied by the theory not the statistic • Yet if some dynamic series of Xs explains more of the dynamics of a set of Ys, then we may say that X Granger-causes Y • The test statistic is a block-F test
Vector Autoregression models
• Structural Equation Models (SEQ) models impose
a priori
restrictions on the theoretical exposition of the theory • VAR models seek to implement tests of theory with fewer restriction.
• They represent a tradeoff between accuracy of causal inference and quantitative precision.
• They better characterize uncertainty and model dynamics.
The VAR Model
• Vector Autoregression is not a statistical technique – It is a design • The VAR Model is:
y t
A
(
L
)
y t
1
u t where A
(
L
)
A
1
A
2
L
A
3
L
2 ...
Vector Autoregression
• Vector Autoregression Models (VARs) are best seen in contrast to Simultaneous Equation Models (SEQs) • SEQ models involve a set of endogenous variables regressed on a set of exogenous variables, with appropriate lag structures supplied for dynamic processes, including simultaneity.
An SEQ Model
• For Instance:
Y
1
t Y
2
t
B B
4 0
B
1
X
1
t B
5
X
3
t
B
2
X
2
t B
6
X
4
t
B
3
Y
2
t B
7
Y
1
t
1 • Note that endogenous variables of one equation may be exogenous in another.
• The lag structure is specifically articulated • The causal nature of the model is explicit – it is a product of the theoretical specification of the model
A VAR
• The equivalent VAR would look like this:
Y
1
t X
1
t
f
(
X
1
t
,
X
1
t
1 ,..
X
2
t
,
X
2
t
1 ...)
f
(
X
2
t
,
X
2
t
1 ,..
X
3
t
,
X
2
t
1 ....,
Y
1
t
,
Y
1
t
1 ..)
etc
..
• The VAR model does not specify specific causation, nor lag structures.