Stata command for maximum likelihood estimation of time trend equation

[Alex Mai]

Dear Statalists,

I would like to ask for the stata command for the maximum likelihood estimation based on first-order autoregressive error terms AR(1) for a simple log-linear trend equation, ln(y)=a+bt+u, where a is the constant, t is the time variable, and u is error term.

Moreover, I would also like to ask how to implement the maximum likelihood procedure to correct first order autocorrelation in the above simple log-linear trend equation (if it is estimated by OLS).

Thank you very much!

[Alex Mai]

Any suggestions would be appreciated! thank you!

[Steve Samuels]

I'm far from knowledgeable in this area, but I for your two problems I suggest an ARMAX model ( with time as covariate) with arima for the first and newey for your second. arima is based on maximum likelihood, but newey is not.

[Alex Mai]

I'm far from knowledgeable in this area, but I for your two problems I suggest an ARMAX model ( with time as covariate) with arima for the first and newey for your second. arima is based on maximum likelihood, but newey is not.

Dear Steve,

Thank you very much! May I ask the advantage of maximum likelihood estimation (-arima-) over OLS estimation in estimating such simple univariate time trend equation(i.e. ln(y)=a+bt+u)? In the economics literature, OLS seems to be more commonly used, and MLE is just sometimes used as a supplement.

[Steve Samuels]

arima will estimate the autocorrelation(s) and the estimated residual SD, under the assumption of normal errors; . newey will give the OLS coefficients and standard errors corrected for autocorrelations (you specify the number of lags), but it will not estimated a corrected residual SD and does not assume normality, I believe. . I suggest that you try both. But I have to ask: 1) What is the evidence that the log-linear model is correct? 2) how do you know that there is just one lagged correlation? That's an assumption that can be tested. At this point, I've reached the limit of my knowledge, so please start another thread if you want to pursue questions about these time-series commands.

[Alex Mai]

arima will estimate the autocorrelation(s) and the estimated residual SD, under the assumption of normal errors; . newey will give the OLS coefficients and standard errors corrected for autocorrelations (you specify the number of lags), but it will not estimated a corrected residual SD and does not assume normality, I believe. . I suggest that you try both. But I have to ask: 1) What is the evidence that the log-linear model is correct? 2) how do you know that there is just one lagged correlation? That's an assumption that can be tested. At this point, I've reached the limit of my knowledge, so please start another thread if you want to pursue questions about these time-series commands.

Dear Steve,

Thank you very much once more! Sorry for taking your time, but may I just ask one more very basic issue about -arima-? Do you think that stationarity does not matter for -arima-?

-arima- will firstly difference the original series, and then use the differenced series. I have tested the stationarity of the US Wholesale Price Index (WPI) used in the Stata manual of -arima-. Neither the original WPI series nor the first-differenced WPI is stationary. But the Stata manual still uses it (in the first-differenced form) to demonstrate -arima-.

So does it mean that I can directly apply -arima- to non-stationary data?

Many thanks!

[Steve Samuels]

Sorry, I don't feel qualified to answer further TS questions.