Tutorial 8 - Time Series

Question 1

What is stationarity?

Answer 1

Stationarity means that the statistical properties of a time series process (such as mean, variance, covariance) do not change systematically over time.

Question 2

How does stationarity relate to OLS?

Answer 2

• It plays an important role for consistency of the OLS estimator. Therefore we are interested in testing whether variables are stationary.
• If we find evidence that a variable is nonstationary, we have to transform it in a way that it becomes stationary before we can use it in an OLS regression.

Question 3

What are the requirements for a time series y_t to be weakly stationary?

Answer 3

• The last criterion implies that e.g. the correlation of y₃ and y₅ is the same as the correlation between y₁₀ and y₁₂ (only the absolute distance matters, not the starting position).
• We can also state that the Cov(y; y_t+h) depends only on h, not on t.

Question 4

What are the requirements for a time series y_t to be strictly stationary?

Answer 4

• if, for any collection of time indices t₁, …, t_n, the joint distribution of y_t1, …, y_tn is the same as the joint distribution of y_t1+h, …, y_tn+h.
• This assumption is stronger than weak stationarity because it makes a statement about the whole distribution of y_t, not just the first/second moments (mean/variance/covariance).
• If y_t is strictly stationary and has a nite second moment (E(y_t²) < ∞), y_t is also weakly stationary. The converse does not hold in general.

Question 5

What are the properties of White Noise?

Answer 5

A series t is called White Noise if

• E(ϵ_t) = 0,
• Var(ϵ_t) = σ², and
• Cov(ϵ_t, ϵ_s) = 0 for all s ≠ t

The ϵ_t are i.i.d.

Obviously, this process is weakly stationary and we write:

Question 6

What is a deterministic time trend?

Answer 6

• is NOT stationary, because E(y_t) = βt (the mean depends systematically on time).
• Var(y_t) = σ² and Cov(y_t, y_t+h) = 0.

Question 7

How can you deal with a deterministic time trend?

Answer 7

• “de-trending” the time series (removing the average time trend) makes it stationary:
  
  • y_t − avgy = β_t + ϵ_t - β_t = ϵ_t
• as the remainder, ϵt , is stationary by construction (White Noise).
• In practise, we can either perform the detrending by hand or simply control for the time trend in the regression. (remember Frisch-Waugh theorem!!!)

Question 8

What is a Stochastic time trend (“Random Walk”)?

Answer 8

• The current value is the past value plus a random shock in the current period.
• A random walk is nonstationary.

Question 9

How can you see that the stochastic time trend (random walk) is nonstationary?

Answer 9

• We can write the series as the starting value y₀ plus the history of all previous shocks up to the present period t:
  
  • y_t = y₀ + ϵ₁ + ϵ₂ + … + ϵ_t
• Then, for the mean we have (note that the ϵ_t have mean zero):
  
  • E(y_t) = E(y₀)
• Variance [Var(y₀) = 0] & Covariance see below