11.4 Forecast

Foreceast: out-of-sample

Before we are able to forecast, we must populate the exogenous variables over the entire forecast horizon before solving our model. 添加数据

Solving our model: means obtain forecast from our model.

Procedure:

  1. Estimate the model

    arima y2 x3 y1, ar(1) ma(1)

  2. Store the estimation results using estimate store

    estimate store myarima

  3. Create a forecast model using forecast create. This initialize a new model; we will call the model mymodel.

    forecast create mymodel

    The name you give the model mainly controls how output from forecast commands is labeled. More importantly, forecast create creates the internal data structures Stata uses to keep track of your model.

  4. Add all equations to the model you just created using forecast estimates.

    The following command adds the stored estimation results in myarima to the current model mymodel.

    forecast estimates myarima

  5. Compute dynamic forecasts from 2012 to 2024

    forecast solve, begin(2012) end(2024)

Creates a new forecast model

forecast create [ name ] [ , replace ]

The forecast create command creates a new forecast model in Stata. You must create a model before you can add equations or solve it. You can have only one model in memory at a time.

You may optionally specify a name for your model. That name will appear in the output produced by the various forecast subcommands.

replace clear the existing model from memory before creating name. By default, forecast create issues an error message if another model is already in memory.

Note that you can add multiple equations to a forecast model.

Add equations/identifies

Add estimation results to a forecast model currently in memory.

forecast estimates modelname [, options]

modelname is the name of a stored estimation result being added; it is generated by estimates store modelname.

Options:

  • predict(p_options): call predict using p_options

  • names(newnamelist[ , replace]): use newnamelist for the names of left-hand-side (LHS) variables in the estimation result being added, i.e., modelname.

    forecast estimates creates a new variable in the dataset for each element of namelist.

    You MUST use this option of any of the LHS variables contains time series operators, e.g., D., L..

    If a variable of the same name already exists in your dataset, forecast estimates exits with an error unless you specify the replace option, in which case existing variables are overwritten.

// use example
forecast estimates myestimates

Add estimation results stored in myestimates to the forecast model currently in memory.

Add an Identity to a forecast Model

forecast identity varname = exp

An identity is a nonstochastic equation that expresses an endogenous variable in the model as a function of other variables in the model. Identities often describe the behavior of endogenous variables that are based on accounting identities or adding-up conditions.

// Add an identity to the forecast that states that y3 is the sum of y1 and y2
forecast identity y3=y1+y2
// create new variable newy before adding it to the forecast
forecast identity newy=y1+y2, generate

The difference is that if the LHS variable does not exist, you need to specify the option gen.

Ex. We have a model using annual data and want to assume that our population variable pop grows at 0.75% per year. Then we can declare endogenous variable pop by using forecast identity:

forecast identity pop = 1.0075*L.pop

Typically, you use forecast identity to define the relationship that determines an endogenous variable that is already in your dataset.

The generate option of forecast identity is useful when you wish to use a transformation of one or more endogenous variables as a right-hand-side variable in a stochastic equation that describes another endogenous variable.

Add equations that you obtained elsewhere to your model

Up untill now, we have been using model output from Stata to add equations to a forecast model, i.e., using forecast estimates.

You use forecast coefvector to add endogenous variables to your model that are defined by linear equations.

Common use scenarios of forecast coefvector:

  • Sometimes, you might see the estimated coefficients for an equation in an article and want to add that equation to your model. In this case, forecast coefvector allows you to add equations that are stored as coefficient vectors to a forecast model.

  • User-written estimators that do not implement a predict command can also be included in forecast models via forecast coefvector.

  • forecast coefvector can also be useful insituations where you want to simulate time-series data.

forecast coefvector cname [, options ]

cname is a Stata matrix with one row. It defines the linear equations, which are stored in a coefficient (parameter) vector.

Options:

  • variance(vname): specify parameter variance matrix of the estimated parameters.

    This option only has an effect if you specify the simulate() option when calling forecast solve and request sim_technique’s betas or residuals.

  • errorvariance(ename): specify additive error term with variance matrix ename, where ename is the name of s Stata matrix. The number of rows and columns in ename must match the number of equations represented by coefficient vector cname.

    This option only has an effect if you specify the simulate() option when calling forecast solve and request sim_technique’s betas or residuals.

  • names(namelist[ , replace ]): instructs forecast coefvector to use namelist as the names of the left-hand-side variables in the coefficient vector being added. By default, forecast coefvector uses the equation names on the column stripe of cname.

    You must use this option if any of the equation names stored with cname contains time-series operators.

You use forecast coefvector to add endogenous variables to your model that are defined by linear equations, where the linear equations are stored in a coefficient (parameter) vector.

// Incorporate coefficient vector of the endogenous equation of y to be used by forecast solve
forecast coefvector y

Ex. We want to add the following eqns to a forecast model. \[ \begin{split} x_t &= 0.2 + 0.3 x_{t-1} - 0.8 z_t \\ z_t &= 0.1 + 0.7 z_{t-1} + 0.3 x_t - 0.2 x_{t-1} \end{split} \]

We first define the coefficient vector eqvector.

// define a row vector
matrix eqvector = (0.2, 0.3, -0.8, 0.1, 0.7, 0.3, -0.2)
// add equation names and variale names
// equation names are before the colon
// variable names are after the colon
matrix coleq eqvector = x:_cons x:L.x x:y y:_cons y:L.y y:x y:L.x
matrix list eqvector

We could then add the coefficient vector to a forecast model.

forecast create
forecast coefvector y

forecast adjust adjusts a variable by add factoring, replacing, etc.

forecast adjust varname = exp [if] [in]

varname is the name of the endogenous variable that has been previously added to the model using forecast estimates or forecast coefvector.

forecast adjust specifies an adjustment to be applied to an endogenous variable in the model. Adjustments are typically used to produce alternative forecast scenarios or to incorporate outside information into a model.

// Adjust the endogenous variable y in forecast to account for the variable shock in 1990
forecast adjust y = y + shock if year==1990

Solve the foreceast

forecast solve computes static or dynamic forecasts based on the model currently in memory. Before you can solve a model, you must first create a new model using forecast create and add equations and variables to it using forecast estimates, forecast coefvector, or forecast identity.

forecast solve [, { prefix(string) | suffix(string) } options ]

Options:

  • prefix(string) and suffix(string) specify prefix/suffix for forecast variables.

    You may specify prefix() or suffix() but NOT both.

    By default, forecast values will be prefixed by f_.

  • begin(time_constant) and end(time_constant) specify period to begin/end forecasting

  • periods(#) specify number of periods to forecast

  • static produce static forecasts instead of dynamic forecasts

    Actual values of variables are used wherever lagged values of the endogenous variables appear in the model. Static forecasts are also called one-step-ahead forecasts.

    By default, dynamic forecasts are produced, which use the forecast values of variables wherever lagged values of the endogenous variables appear in the model.

  • actuals use actual values if available instead of forecasts

    actuals specifies how nonmissing values of endogenous variables in the forecast horizon are treated. By default, nonmissing values are ignored, and forecasts are produced for all endogenous variables. When you specify actuals, forecast sets the forecast values equal to the actual values if they are nonmissing. The forecasts for the other endogenous variables are then conditional on the known values of the endogenous variables with nonmissing data.

  • log(log_level) loglevel takes on one of the following values

    • on: default, provides an iteration log showing the current panel and period for which the model is being solved as well as a sequence of dots for each period indicating the number of iterations.
    • off: suppress the iteration log.
    • detail: a detailed iteration log including the current values of the convergence criteria for each period in each panel (in the case of panel data) for which the model is being solved.
    • brief: produces an iteration log showing the current panel being solved but does not show which period within the current panel is being solved.

  • simulate(sim_technique, sim_statistic sim_options) allows you to simulate your model to obtain measures of uncertainty surrounding the point forecasts produced by the model. Simulating a model involves repeatedly solving the model, each time accounting for the uncertainty associated with the error terms and the estimated coefficient vectors.

    • sim_technique can be betas, errors, or residuals.

      • betas: draw multivariate-normal parameter vectors
      • errors: draw additive errors from multivariate normal distribution
      • residuals: draw additive residuals based on static forecast errors

    • sim_statistic specifies a summary statistic to summarize the forecasts over all the simulations.

    statistic(statistic, { prefix(string) | suffix(string) })

    statistic can be mean, variance, or stddev. You may specify either the prefix or the suffix that will be used to name the variables that will contain the requested statistic.

    • sim_options includes
      • saving(filename, …) save results to file
      • nodots suppress replication dots. By default, one dot character is displayed for each successful replication. If during a replication convergence is not achieved, forecast solve exits with an error message.
      • reps(#) request that forecast solve perform # replications; default is reps(50)

Use example: forecast a panel

\[ \%\Delta \text{dim}_{it} = \beta_0 + \beta_1 \ln(\text{starts}_{it}) + \beta_2 \text{rgspgrowth}_{it} + \beta_3 \text{unrate}_{it} + u_{i} + \varepsilon_{it} \]

\(u_{i}\) refers to individual fixed effects.

When we make forecasts for any individual panel, we may want to include it in our forecasts. This can be achieved by using forecast adjust.

use https://www.stata-press.com/data/r19/statehardware, clear
generate lndim = ln(dim)
generate lnstarts = ln(starts)
quietly xtreg D.lndim lnstarts rgspgrowth unrate if qdate <= tq(2009q4), fe
predict dlndim_u, u  /* obtain individual fixed effects */
estimates store dim  /* store estimation results */

With enough observations, we can have more confidence in the estimated panel-specific errors. If we are willing to assume that we have decent estimates of the panel-specific errors and that those panel-level effects will remain constant over the forecast horizon, then we can incorporate them into our forecasts.

Because predict only provided us with estimates of the panel-level effects for the estimation sample, we need to extend them into the forecast horizon. An easy way to do that is to use egen to create a new set of variables:

// extend panel fixed effects to the forecast horizon
by state: egen dlndim_u2 = mean(dlndim_u)

We can use forecast adjust to incorporate these terms into our forecasts.

The following commands define our forecast model, including the estimated panel-specific terms:

/* create forecast model */
forecast create statemodel, replace   
/* add equations, rename the endog variable, D.lndim, to be forecasted as dlndim */
/* since the original endog variable name includes a time series operator
   it is required to name, otherwise will return error */
forecast estimates dim, name(dlndim)  
/* add state fixed effects */
forecast adjust dlndim = dlndim + dlndim_u2

  • Note that our dependent variable contains a time series operator, we must use name(dlndim) option of forecast estimates to specify a valid name for the endogenous variable being added.

  • dlndim stands for the first difference of the logarithm of dim. We are interested in the level of dim, so we need to back out dim from dlndim.

    → We use forecast identity to obtain the actual dim variable.

// reverse first difference, note that you refer to the endog var using the new name, dlndim, now
forecast identity lndim = L.lndim + dlndim
// reverse natural logarithm
forecast identity dim = exp(lndim)

We used forecast adjust to perform our adjustment to dlndim immediately after we added those estimation results so that we would not forget to do so. However, we could specify the adjustment at any time.

Regardless of when you specify an adjustment, forecast solve performs those adjustments immediately after the variable being adjusted is computed.

Finally we can solve the model. Here we obtain dynamic forecasts beginning in the first quarter of 2010:

forecast solve, begin(tq(2010q1)) log(off)