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Aksakal
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You can start with assuming that your observed variable is obtained from the true value as $$y_t = \theta_0 x_t + \theta_1 x_{t-1} + e_t$$

It would help to know what is the process of the underlying variable, suppose it's $$x_t = \beta_0 + \beta_1 x_{t-1} + u_t$$

where $e_t,u_t$ are errors. If these equations make sense to you then, you can estimate them using Kalman filter, see example here.

Next, you test whether $\theta_0+\theta_1=1$, if it holds statistically, then maybe your specification holds, so you can proceed with a constrained fit.

You have to set the expectations though: smoothing leads to data loss, generally. So, you can't reproduce the original series exactly. That's why using Kalman filter we had to make an assumption about the observed and true processes, i.e. we needed to inject some outside data to compensate for lost data (from smoothing) in order to recover the true series.

You can start with assuming that your observed variable is obtained from the true value as $$y_t = \theta_0 x_t + \theta_1 x_{t-1} + e_t$$

It would help to know what is the process of the underlying variable, suppose it's $$x_t = \beta_0 + \beta_1 x_{t-1} + u_t$$

where $e_t,u_t$ are errors. If these equations make sense to you then, you can estimate them using Kalman filter, see example here.

You have to set the expectations though: smoothing leads to data loss, generally. So, you can't reproduce the original series exactly. That's why using Kalman filter we had to make an assumption about the observed and true processes, i.e. we needed to inject some outside data to compensate for lost data (from smoothing) in order to recover the true series.

You can start with assuming that your observed variable is obtained from the true value as $$y_t = \theta_0 x_t + \theta_1 x_{t-1} + e_t$$

It would help to know what is the process of the underlying variable, suppose it's $$x_t = \beta_0 + \beta_1 x_{t-1} + u_t$$

where $e_t,u_t$ are errors. If these equations make sense to you then, you can estimate them using Kalman filter, see example here.

Next, you test whether $\theta_0+\theta_1=1$, if it holds statistically, then maybe your specification holds, so you can proceed with a constrained fit.

You have to set the expectations though: smoothing leads to data loss, generally. So, you can't reproduce the original series exactly. That's why using Kalman filter we had to make an assumption about the observed and true processes, i.e. we needed to inject some outside data to compensate for lost data (from smoothing) in order to recover the true series.

Source Link
Aksakal
  • 55.3k
  • 5
  • 87
  • 176

You can start with assuming that your observed variable is obtained from the true value as $$y_t = \theta_0 x_t + \theta_1 x_{t-1} + e_t$$

It would help to know what is the process of the underlying variable, suppose it's $$x_t = \beta_0 + \beta_1 x_{t-1} + u_t$$

where $e_t,u_t$ are errors. If these equations make sense to you then, you can estimate them using Kalman filter, see example here.

You have to set the expectations though: smoothing leads to data loss, generally. So, you can't reproduce the original series exactly. That's why using Kalman filter we had to make an assumption about the observed and true processes, i.e. we needed to inject some outside data to compensate for lost data (from smoothing) in order to recover the true series.