Autocorrelation Flashcards
What is autocorrelation
The values of the disturbance term u are not independent of each other
What has to to hold to prevent autocorrelation
Cov(ui, uj ) = 0
How many lags to use in time series data
Trial and error, but if they want a full year report then quarterly would have four lags
yt = β0 + β1xt + β2xt−1 + β3xt−2 + β4xt−3 + β5xt−4 + ut
Why might we use lags of Y as a lagged dependent variable
If inertia effects are important - if this year is influenced by what you did last year
yt = β0 + β1xt + β2xt−1 + β3yt−1 + ut
Types of autocorrelation
Autoregressive of order 1
Spatial autocorrelation
What is AR (1)
Where the disturbance term in an observation is related to the disturbance term in the observation before
What does ut = for AR (1)
put-1 + et
Negative autocorrelation on a graph
Successive values tend to have different signs.
positive values followed by negative ones
Positive autocorrelation on a graph
Positive values followed by positive and negative followed by negative
What are the consequences of autocorrelation : Coefficients, standard errors, standard errors for regression coefficients
Does not bias the estimated coefficients
However they will be inefficient
Standard errors of the regression coefficients are estimated wrongly ( for AR(1) they are underestimated )
Any f and t tests are invalid
Spatial autocorrelation
systematic pattern in the spatial distribution of a variable
Difference between positive and negative spatial autocorrelation
Negative - Neighboring areas are unlike
Positive - neighboring areas are more alike
Ways to test autocorrelation
Durbin-Watson test
With lagged dependent variable
Durban watson formula
d =
t=1 uˆ2t
Scores of durban watson tests
d=2 no autocorrelation
d=0 severe positive
d=4 severe negative
Draw out Durban Watson test between 0-4
with dl du dcrit etc
If d lies between dL and dU
we fail to reject the null
If d lies between dU and 2
we would fail to reject the null
If d is less than dL
Reject the null and conclude there is autocorrelation
Reading a Durbin Watson test
N - number of observations
K - number of explanatory variables
Why doesn’t a Durban Watson test work for Autocorrelation with a lagged dependent variable. Also what to do now?
It is biased towards the value 2 so therefore not of use. Use Durbin’s h test
Durbin’s h test
h = ˆρ n
1 − ns2↓βˆyt−1
everything inside a square root apart from the p hat
Three items required for test statistic. And how they are calculated
Number of observations in the regression n
An estimate of the variance of the coefficient of the lagged dependent variable - will be some letter t-1
An estimate for p. P hat = 1-o.5d
In large samples what does d =
2-2p
Ho p=0
no autocorrelation
h statistic distribution and mean and variance
Normal
Mean = 0
Variance = 1
Therefore can use normal distribution table to test
How to eliminate AR(1) autocorrelation
Use generalised least squares (GLS)
Cochrane-Orcutt Iterative method - used before computers
Method of GLS
Multiply regression through by p, then put all variables to t-1
Subtract the new equation from the first
This reduces the disturbance term to Et
Cochrane-Orcutt iterative process
Multiply regression through by p, then put all variable to t-1
Define new key terms to make the regression linear.
yt hat = yt-py↓t-1
xt hat = xt-px↓t-1
B’0 = B0(1-P)
1. Regress yt on xt using OLS.
2. Calculate uˆt = yt − βˆ0 − βˆ1x and regress uˆt on uˆt−1 to obtain an estimate of ρ.
3. Calculate y˜t and x˜t and regress y˜t on x˜t to obtain revised estimates
of β0 and β1.
4. Return to (2) and continue until convergence – i.e. get the same estimates for β0 and β1 that we obtained in the previous iteration.
Common factor test
X^2 = n * ln(SSRr/SSRu) n = number of observations - sample number - 1
Causes of autocorrelation
- Something in the disturbance term that has a persistent effect – AR(1) autocorrelation.
- The omission of lagged variables that should be in the equation
