 # javlacalle

http://jalobe.com

 Jul 6 awarded Necromancer Jun 25 comment Is there a way to allow seasonality in regression coefficients?@Frank If 11 colums are used in $SDB$, then you would in principle include a constant $\alpha$ (a column of ones); otherwise, the residuals may not be zero on average: $G_t = \alpha + Z_t + \beta_{0,s} SD_t + \beta_{1,s} SDB_t$. At the 12-th season (the one left out), the expected value of $G_t$ is $\alpha+\beta_{1,12}SDB_t$. The coefficients $\beta_{0,s}$, $s=1,\dots,11$ are interpreted as changes with respect to the estimate of $\alpha$. Jun 23 comment Is there a way to allow seasonality in regression coefficients?@Frank The intercept is set to zero for the season that is left out. The coefficients of the intercepts related to the remaining coefficients are interpreted as a change with respect to the average value of the deleted season (which is no necessarily zero, but the value determined by the coefficients and values of the remaining variables at that season). Jun 21 awarded Yearling Jun 21 awarded Yearling Jun 18 awarded Yearling Jun 18 awarded Yearling Jun 3 awarded Good Answer Apr 24 awarded Revival Apr 16 comment How do I detrend time series?@student1 As the consequences of omitting a unit root when it is present are more dangerous than considering the presence of a unit root when the process is actually stationary, we may give preference to have the chance to reject the hypothesis of stationarity when there is a unit root, rather than reject a unit root when the process is stationary. The sequence KPSS-ADF is, in this sense, a safer approach. Apr 16 comment How do I detrend time series?@student1 It's not clear to me whether it is better to apply the sequencey ADF-KPSS or KPSS-ADF. As the test statistics are not independent of each other, it is not straightforward to get the level and power of the sequence of the tests. I don't know about a theoretical work on it. Some simulations can be performed to explore this issues. For illustration, you may check the exercises in pp. 42-44 in this document. Mar 26 awarded Nice Answer Feb 19 awarded Enlightened Jan 5 comment Polynomial regression for missing value imputationYou may be interested in this post. Jan 5 comment ARMA process forecasts and maximum likelihood parametersI haven't seen this approach to obtain forecasts before, so I cannot say much about this part of your question. Just some thoughts: 1) what you say about forecasting without parameters seems interesting in this approach, but isn't it $\mathbf{a}$ a set of parameters anyway? 2) If I am correct, $Y$ is the observed time series and $X$ are explanatory variables (e.g., lags of $Y$ and lags of the disturbance term $\varepsilon$). If this is correct, it may not be straightforward to calculate $E(XX^T)$ because $\varepsilon_t$ are not observed (unlike the lags of $Y$). Jan 4 comment ARMA process forecasts and maximum likelihood parametersOnce the algorithm that combines the Kalman filter and the Newton method converges to a set of parameter values, the data can be extrapolated by means of the Kalman filter (maybe this is what you see related to your last comment). In practice, it may be better to use the Kalman smoother to obtain forecasts (employs information on the whole data set). Jan 4 comment ARMA process forecasts and maximum likelihood parametersMaximum likelihood has in general better properties than the method of moments, related post. The method of moments is nonetheless a helpful procedure to get initial parameter estimates from which to start the search of ML estimates. Jan 4 comment ARMA process forecasts and maximum likelihood parametersBut why not estimate [second moments] from data? Calculating the sample autocovariances from the data is a natural alternative. In fact, the theory of ARMA models establishes a mapping between the theoretical autocovariances and the parameters of the model. This approach is known as the method of moments. For AR models it amounts to the Yule-Walker system of equations. When MA is present, the system of equations is less simple to solve. Jan 4 comment ARMA process forecasts and maximum likelihood parametersBut how are parameters of ARMA for maximum likelihood found? The likelihood function is maximised with respect to the parameters of the model (AR and MA coefficients). To do so, an optimisation algorithm and the Kalman filter can be combined as follows: The Kalman filter is used to compute the value of the likelihood function for a given set of parameters values; it is passed as argument to an optimisation algorithm, eg. Newton method. As a result, a set of parameter values that maximises the likelihood function are obtained and used to compute forecasts upon the given model. Jan 4 comment ARMA process forecasts and maximum likelihood parametersWhy do we even need likelihood at all? Innovations and observations are not enough, the parameters of the model ($\phi\,,\theta$) needs to be estimated. ARMA models with the same orders, eg. ARMA(2,2), but with different values of the AR and MA coefficients do not involve the same features (period of cycles, persistency,...). This is where the likelihood function comes into play; it measures the likelihood or plausibility of a set of parameters values given the observed data. So the goal is to chose those parameter values that maximise the likelihood function.