Non-stationarity: trends, random walks, unit roots and time-varying volatility
Why do we care about sta onarity?
- OLS assump ons imply sta onarity
- Not sa sfied; OLS gives biased or inconsistent es mates
- Regressing a non-sta onary variable on a completely unrelated other non-sta onary variable, can
obtain spuriously (false/fake) good results with a high R^2.
Simula on of spurious regression (regressing two independent (non-sta onary) variables on each other many
mes), results in:
- In many cases high R^2
- In many cases very high T-stat
Two types of non-sta onarity
1. Most common in finance: Random walk model with dri (stochas c trend):
Could be generalised to an Explosive case:
Typically, the explosive case is ignored and we use . Because does not describe many
data series in finance and has a counterintui ve property (current shock will have increasingly large
influence in the future).
2. Determinis c trend process:
Where sa sfies the OLS assump ons in both cases (white noise). Constant variance, not correlated over
me.
, Impact of shocks for sta onary and non-sta onary series, 3 cases:
1. Sta onary.
Recent shocks ma er a lot and far in past ma er a li le. Shocks gradually die away.
2. Random walk model with dri .
Shocks persist and never die away. So just an infinite sum of past shocks plus some star ng value of Y0.
3. Explosive case.
Shocks become more influen al as me goes on.
Remove sta onarity from random walk (1) process by differencing:
Example:
Remove sta onarity from determinis c trends (2) by adding me as an explanatory variable and then add
(sta onary) explanatory variables.
Various stochas c processes graphed:
Why do we care about sta onarity?
- OLS assump ons imply sta onarity
- Not sa sfied; OLS gives biased or inconsistent es mates
- Regressing a non-sta onary variable on a completely unrelated other non-sta onary variable, can
obtain spuriously (false/fake) good results with a high R^2.
Simula on of spurious regression (regressing two independent (non-sta onary) variables on each other many
mes), results in:
- In many cases high R^2
- In many cases very high T-stat
Two types of non-sta onarity
1. Most common in finance: Random walk model with dri (stochas c trend):
Could be generalised to an Explosive case:
Typically, the explosive case is ignored and we use . Because does not describe many
data series in finance and has a counterintui ve property (current shock will have increasingly large
influence in the future).
2. Determinis c trend process:
Where sa sfies the OLS assump ons in both cases (white noise). Constant variance, not correlated over
me.
, Impact of shocks for sta onary and non-sta onary series, 3 cases:
1. Sta onary.
Recent shocks ma er a lot and far in past ma er a li le. Shocks gradually die away.
2. Random walk model with dri .
Shocks persist and never die away. So just an infinite sum of past shocks plus some star ng value of Y0.
3. Explosive case.
Shocks become more influen al as me goes on.
Remove sta onarity from random walk (1) process by differencing:
Example:
Remove sta onarity from determinis c trends (2) by adding me as an explanatory variable and then add
(sta onary) explanatory variables.
Various stochas c processes graphed:
