Lecture 7

Question 1

Taking the natural logarithm is the inverse of taking an exponential. Please outline three reasons why logarithmic transformation might be useful

Answer 1

-It can help rescale data so the variance is more constant

-It can help make a positively skewed distribution closer to a normal distribution

-Taking a logarithm can make a nonlinear, multiplicative relationship between variables into a linear, additive one.

Question 2

Please describe the autocorrelation function and the partial autocorrelation function

Answer 2

Whne the linear dependence between yt and its past values (yt-1, yt-2 etc.) is of interest, the concept of correlation is generalized to autocorrelation

PACF measures the correlation between a specific observation ‘k’ and the current observation ‘yt’, controlling ‘ignoring’ all lags in between

Question 3

What are univariate time-series models?

Answer 3

We call it univariate because we use only look at time-series models that use the past values of the dependent variable itself. Only look at Y, not X.

Question 4

Please outline the autoregressive model

Answer 4

We use the autoregressive (AR) model to test for the dependence of the yt term on its previous values of itself.

Yt depends on its values in previous periods plus a white noise disturbance term

Question 5

Consider the simple AR model:
𝑦𝑡 = 𝜙𝑦𝑡−1 + 𝑢t

Please describe the meaning of:

𝜙 = 1
𝜙 < 1
𝜙 > 1

Answer 5

If 𝜙 = 1:
We see a unit-root meaning yt is non-stationary and follows a random walk process. This means it is entirely unpredictable.

If 𝜙<1:
Then yt is stationary and we see that yt exhibits mean-reversion. Not a good description of price, but a good descriptor of returns.

If 𝜙>1:
Then yt follows an explosive process. This is rare for financial time series

Question 6

What is stationarity?

Please also outline the difference between strict stationarity and weakly/covariance stationarity

Answer 6

Not only a desirable property of an AR model but also the foundation of time series analysis.

• A flat time series is likely to be stationary

Strict stationarity:
This requires that the joint distribution of its values remain the same as time progresses. This is quite rare.

Weakly/covariance:
The mean and variance of its values are ‘time invariant’ (it does not change with time) and the covariance between xt and xt+h depends only on the distance between the two terms (h) and not on the location of the initial time period (t)

Question 7

How do we test for stationarity?

Answer 7

We perform the unit root test.

H0: δ = 0, series y has a unit rootm (is non-stationary)
H1: δ < 0, series y is stationary