Covariance stationary time series

[cfageist]

Hi,

I understand that if both the intercept and slope coefficient in a time series do not differ significantly from 0 then it is a covariance stationary time series but how can we make valid inferences if both are not statistically significant? Does the fact that the time series is covariance stationary take precedence over the fact that the regression coefficients are statistically insignificant? Would we still be able to make proper statistical inferences?

thanks!

[S2000magician]

b₁ doesn’t need to be zero. As long as |b₁| < 1 and b₀ = 0 it’s covariance stationary.

[cfageist]

thanks, how about if you have a random walk? If you first difference the time series, should your output show statistical insignificance for both regression coefficients? In that case would inferences still be valid?

[S2000magician]

A random walk is not covariance stationary because b₁ = 1.

[cfageist]

I know but if it’s first differenced, the expectation is that b₀ and b₁ equal 0, right?

[S2000magician]

Yes.

To what inferences are you referring?

[cfageist]

I am referring to the inferences we can make based on the regression coefficients (b₁ and b₀). If under the assumption that they should be 0 for first differenced time series, wouldn’t that make them incapable of estimating the dependent variable? In the reading on linear regression, the first thing we test is whether the regression coefficients are different from 0, if they are not then the independent variables may not have explanatory power. I guess my question boils down to: can we make valid estimations if both regression coefficients are 0 in a time series?

[S2000magician]

If in the first difference equation b₁ can be zero, then in the original equation b₁ can be 1. This means that the original equation might not be covariance stationary.

[gargijain]

I am confused from where this is coming?

I know but if it’s first differenced, the expectation is that b₀ and b₁ equal 0, right?

[cfageist]

Check in the time series reading. It is in the random walk section.

[gargijain]

Got it, thanks…

Check in the time series reading. It is in the random walk section.

[gargijain]

What I understand is that in a random walk, we cannot use b0 and b1 to predict, thats why we first difference and use the error term to predict. So, to answer your question, b1 and b0 will not be used to predict. In Random walk, we will add the error term to the previous value and predict the next.

Magician, pls correct me if I am wrong…

@cfageist

I am referring to the inferences we can make based on the regression coefficients (b₁ and b₀). If under the assumption that they should be 0 for first differenced time series, wouldn’t that make them incapable of estimating the dependent variable? In the reading on linear regression, the first thing we test is whether the regression coefficients are different from 0, if they are not then the independent variables may not have explanatory power. I guess my question boils down to: can we make valid estimations if both regression coefficients are 0 in a time series?

[Harrogath]

@gargijain

What I understand is that in a random walk, we cannot use b0 and b1 to predict, thats why we first difference and use the error term to predict. So, to answer your question, b1 and b0 will not be used to predict. In Random walk, we will add the error term to the previous value and predict the next.

Magician, pls correct me if I am wrong…

@cfageist

I am referring to the inferences we can make based on the regression coefficients (b₁ and b₀). If under the assumption that they should be 0 for first differenced time series, wouldn’t that make them incapable of estimating the dependent variable? In the reading on linear regression, the first thing we test is whether the regression coefficients are different from 0, if they are not then the independent variables may not have explanatory power. I guess my question boils down to: can we make valid estimations if both regression coefficients are 0 in a time series?

Sounds logical for me.