forecast

Forecast responses of Bayesian linear regression model

Syntax

yF = forecast(Mdl,XF)

yF = forecast(Mdl,XF,X,y)

yF = forecast(___,Name,Value)

[yF,YFCov] = forecast(___)

Description

yF = forecast(Mdl,XF) returns numPeriods forecasted responses from the Bayesian linear regression model Mdl given the predictor data in XF, a matrix with numPeriods rows.

To estimate the forecast, forecast uses the mean of the numPeriods-dimensional posterior predictive distribution.

• If Mdl is a joint prior model (returned by bayeslm), then forecast uses only the joint prior distribution and the innovation distribution to form the predictive distribution.

• If Mdl is a posterior model (returned by estimate), then forecast uses the posterior predictive distribution.

NaNs in the data indicate missing values, which forecast removes using list-wise deletion.

yF = forecast(Mdl,XF,X,y) forecasts using the posterior predictive distribution produced or updated by incorporating the predictor data X and corresponding response data y.

• If Mdl is a joint prior model, then forecast produces the posterior predictive distribution by updating the prior model with information about the parameters that it obtains from the data.

• If Mdl is a posterior model, then forecast updates the posteriors with information about the parameters that it obtains from the additional data. The complete data likelihood is composed of the additional data X and y, and the data that created Mdl.

yF = forecast(___,Name,Value) uses any of the input argument combinations in the previous syntaxes and additional options specified by one or more name-value pair arguments. For example, you can specify a value for β or σ² to forecast from the conditional predictive distribution of one parameter, given the specified value of the other parameter.

[yF,YFCov] = forecast(___) also returns the covariance matrix of the numPeriods-dimensional posterior predictive distribution. Standard deviations of the forecasts are the square roots of the diagonal elements.

Examples

Forecast Responses Using Posterior Predictive Distribution

Consider the multiple linear regression model that predicts the US real gross national product (GNPR) using a linear combination of industrial production index (IPI), total employment (E), and real wages (WR).

$${\text{GNPR}}_t={\beta }_0+{\beta }_1{\text{IPI}}_t+{\beta }_2{\text{E}}_t+{\beta }_3{\text{WR}}_t+{\epsilon }_t.$$

For all $$t$$, $${\epsilon }_t$$ is a series of independent Gaussian disturbances with a mean of 0 and variance $${\sigma }^2$$.

Assume these prior distributions:

• $$\beta |{\sigma }^2\sim N_4(M,{\sigma }^{2}V)$$. $$M$$ is a 4-by-1 vector of means, and $$V$$ is a scaled 4-by-4 positive definite covariance matrix.

• $${\sigma }^2\sim IG(A,B)$$. $$A$$ and $$B$$ are the shape and scale, respectively, of an inverse gamma distribution.

These assumptions and the data likelihood imply a normal-inverse-gamma conjugate model.

Create a normal-inverse-gamma conjugate prior model for the linear regression parameters. Specify the number of predictors p and the variable names.

p = 3;
VarNames = ["IPI" "E" "WR"];
PriorMdl = bayeslm(p,'ModelType','conjugate','VarNames',VarNames);

Mdl is a conjugateblm Bayesian linear regression model object representing the prior distribution of the regression coefficients and disturbance variance.

Load the Nelson-Plosser data set. Create variables for the predictor and response data. Hold out the last 10 periods of data from estimation so you can use them to forecast real GNP.

load Data_NelsonPlosser
fhs = 10; % Forecast horizon size
X = DataTable{1:(end - fhs),PriorMdl.VarNames(2:end)};
y = DataTable{1:(end - fhs),'GNPR'};
XF = DataTable{(end - fhs + 1):end,PriorMdl.VarNames(2:end)}; % Future predictor data
yFT = DataTable{(end - fhs + 1):end,'GNPR'};                  % True future responses

Estimate the marginal posterior distributions. Turn off the estimation display.

PosteriorMdl = estimate(PriorMdl,X,y,'Display',false);

PosteriorMdl is a conjugateblm model object that contains the posterior distributions of $$\beta$$ and $${\sigma }^2$$.

Forecast responses by using the posterior predictive distribution and the future predictor data XF. Plot the true values of the response and the forecasted values.

yF = forecast(PosteriorMdl,XF);

figure;
plot(dates,DataTable.GNPR);
hold on
plot(dates((end - fhs + 1):end),yF)
h = gca;
p = patch([dates(end - fhs + 1) dates(end) dates(end) dates(end - fhs + 1)],...
    h.YLim([1,1,2,2]),[0.8 0.8 0.8]);
uistack(p,'bottom');
legend('Forecast Horizon','True GNPR','Forecasted GNPR','Location','NW')
title('Real Gross National Product: 1909 - 1970');
ylabel('rGNP');
xlabel('Year');
hold off

yF is a 10-by-1 vector of future values of real GNP corresponding to the future predictor data.

Estimate the forecast root mean squared error (RMSE).

frmse = sqrt(mean((yF - yFT).^2))

frmse = 25.5397

The forecast RMSE is a relative measure of forecast accuracy. Specifically, you estimate several models using different assumptions. The model with the lowest forecast RMSE is the best-performing model of the ones being compared.

Directly Forecast Observations Using Posterior

Consider the regression model in Forecast Responses Using Posterior Predictive Distribution.

Create a normal-inverse-gamma semiconjugate prior model for the linear regression parameters. Specify the number of predictors p and the names of the regression coefficients.

p = 3;
PriorMdl = bayeslm(p,'ModelType','semiconjugate','VarNames',["IPI" "E" "WR"]);

Load the Nelson-Plosser data set. Create variables for the response and predictor series.

load Data_NelsonPlosser
X = DataTable{:,PriorMdl.VarNames(2:end)};
y = DataTable{:,'GNPR'};

Hold out the last 10 periods of data from estimation so you can use them to forecast real GNP. Turn off the estimation display.

fhs = 10; % Forecast horizon size
X = DataTable{1:(end - fhs),PriorMdl.VarNames(2:end)}; 
y = DataTable{1:(end - fhs),'GNPR'};
XF = DataTable{(end - fhs + 1):end,PriorMdl.VarNames(2:end)}; % Future predictor data
yFT = DataTable{(end - fhs + 1):end,'GNPR'};                  % True future responses

Forecast responses by using the posterior predictive distribution and the future predictor data XF. Specify the in-sample observations X and y (the observations from which MATLAB® composes the posterior).

yF = forecast(PriorMdl,XF,X,y)

yF = 10×1

  491.5404
  518.1725
  539.0625
  566.7594
  597.7005
  633.4666
  644.7270
  672.7937
  693.5321
  678.2268

Forecast Responses After Performing Predictor Variable Selection

Consider the regression model in Forecast Responses Using Posterior Predictive Distribution.

Assume these prior distributions for $\mathit{k}$ = 0,...,3:

• $${\beta }_k|{\sigma }^2,{\gamma }_k={\gamma }_k\sigma \sqrt{V_{k1}}{Z}_1+(1-{\gamma }_k)\sigma \sqrt{V_{k2}}{Z}_2$$, where ${\mathit{Z}}_1$ and ${\mathit{Z}}_2$ are independent, standard normal random variables. Therefore, the coefficients have a Gaussian mixture distribution. Assume all coefficients are conditionally independent, a priori, but they are dependent on the disturbance variance.

• $${\sigma }^2\sim IG(A,B)$$. $$A$$ and $$B$$ are the shape and scale, respectively, of an inverse gamma distribution.

• ${\gamma }_{\mathit{k}}\in \left\{0,1\right\}$and it represents the random variable-inclusion regime variable with a discrete uniform distribution.

Perform stochastic search variable selection (SSVS):

1. Create a Bayesian regression model for SSVS with a conjugate prior for the data likelihood. Use the default settings.

2. Hold out the last 10 periods of data from estimation.

3. Estimate the marginal posterior distributions.

p = 3;
PriorMdl = bayeslm(p,'ModelType','mixconjugate','VarNames',["IPI" "E" "WR"]);

load Data_NelsonPlosser
fhs = 10; % Forecast horizon size
X = DataTable{1:(end - fhs),PriorMdl.VarNames(2:end)};
y = DataTable{1:(end - fhs),'GNPR'};
XF = DataTable{(end - fhs + 1):end,PriorMdl.VarNames(2:end)}; % Future predictor data
yFT = DataTable{(end - fhs + 1):end,'GNPR'};                  % True future responses

rng(1); % For reproducibility
PosteriorMdl = estimate(PriorMdl,X,y,'Display',false);

Forecast responses using the posterior predictive distribution and the future predictor data XF. Plot the true values of the response and the forecasted values.

yF = forecast(PosteriorMdl,XF);

figure;
plot(dates,DataTable.GNPR);
hold on
plot(dates((end - fhs + 1):end),yF)
h = gca;
hp = patch([dates(end - fhs + 1) dates(end) dates(end) dates(end - fhs + 1)],...
    h.YLim([1,1,2,2]),[0.8 0.8 0.8]);
uistack(hp,'bottom');
legend('Forecast Horizon','True GNPR','Forecasted GNPR','Location','NW')
title('Real Gross National Product: 1909 - 1970');
ylabel('rGNP');
xlabel('Year');
hold off

yF is a 10-by-1 vector of future values of real GNP corresponding to the future predictor data.

Estimate the forecast root mean squared error (RMSE).

frmse = sqrt(mean((yF - yFT).^2))

frmse = 18.8470

The forecast RMSE is a relative measure of forecast accuracy. Specifically, you estimate several models using different assumptions. The model with the lowest forecast RMSE is the best-performing model of the ones being compared.

When you perform Bayesian regression with SSVS, a best practice is to tune the hyperparameters. One way to do so is to estimate the forecast RMSE over a grid of hyperparameter values, and choose the value that minimizes the forecast RMSE.

Forecast Responses Using Conditional Posterior Predictive Distribution

Consider the regression model in Forecast Responses Using Posterior Predictive Distribution.

Create a normal-inverse-gamma semiconjugate prior model for the linear regression parameters. Specify the number of predictors p and the names of the regression coefficients.

p = 3;
PriorMdl = bayeslm(p,'ModelType','semiconjugate','VarNames',["IPI" "E" "WR"]);

Load the Nelson-Plosser data set. Create variables for the response and predictor series.

load Data_NelsonPlosser
X = DataTable{:,PriorMdl.VarNames(2:end)};
y = DataTable{:,'GNPR'};

Hold out the last 10 periods of data from estimation so you can use them to forecast real GNP. Turn off the estimation display.

fhs = 10; % Forecast horizon size
X = DataTable{1:(end - fhs),PriorMdl.VarNames(2:end)}; 
y = DataTable{1:(end - fhs),'GNPR'};
XF = DataTable{(end - fhs + 1):end,PriorMdl.VarNames(2:end)}; % Future predictor data
yFT = DataTable{(end - fhs + 1):end,'GNPR'};                  % True future responses

Forecast responses by using the conditional posterior predictive distribution of beta given $${\sigma }^2=2$$ and using the future predictor data XF. Specify the in-sample observations X and y (the observations from which MATLAB® composes the posterior). Plot the true values of the response and the forecasted values.

yF = forecast(PriorMdl,XF,X,y,'Sigma2',2);

figure;
plot(dates,DataTable.GNPR);
hold on
plot(dates((end - fhs + 1):end),yF)
h = gca;
hp = patch([dates(end - fhs + 1) dates(end) dates(end) dates(end - fhs + 1)],...
    h.YLim([1,1,2,2]),[0.8 0.8 0.8])

hp = 
  Patch with properties:

    FaceColor: [0.8000 0.8000 0.8000]
    FaceAlpha: 1
    EdgeColor: [0 0 0]
    LineStyle: '-'
        Faces: [1 2 3 4]
     Vertices: [4x2 double]

  Show all properties

uistack(hp,'bottom');
legend('Forecast Horizon','True GNPR','Forecasted GNPR','Location','NW')
title('Real Gross National Product: 1909 - 1970');
ylabel('rGNP');
xlabel('Year');
hold off

Estimate Forecast Intervals

Consider the regression model in Forecast Responses Using Posterior Predictive Distribution.

Create a normal-inverse-gamma semiconjugate prior model for the linear regression parameters. Specify the number of predictors p and the names of the regression coefficients.

p = 3;
PriorMdl = bayeslm(p,'ModelType','semiconjugate','VarNames',["IPI" "E" "WR"]);

Load the Nelson-Plosser data set. Create variables for the response and predictor series.

load Data_NelsonPlosser
X = DataTable{:,PriorMdl.VarNames(2:end)};
y = DataTable{:,'GNPR'};

Hold out the last 10 periods of data from estimation so you can use them to forecast real GNP. Turn off the estimation display.

fhs = 10; % Forecast horizon size
X = DataTable{1:(end - fhs),PriorMdl.VarNames(2:end)}; 
y = DataTable{1:(end - fhs),'GNPR'};
XF = DataTable{(end - fhs + 1):end,PriorMdl.VarNames(2:end)}; % Future predictor data
yFT = DataTable{(end - fhs + 1):end,'GNPR'};                  % True future responses

Forecast responses and their covariance matrix by using the posterior predictive distribution and the future predictor data XF. Specify the in-sample observations X and y (the observations from which MATLAB® composes the posterior).

[yF,YFCov] = forecast(PriorMdl,XF,X,y);

Because the predictive posterior distribution is not analytical, a reasonable approximation to a set of 95% credible intervals is

$${\underset{}{\overset{ˆ}{y}}}_i\pm z_{0.975}SE({\underset{}{\overset{ˆ}{y}}}_i),$$

for all $$i$$ in the forecast horizon. Estimate 95% credible intervals for the forecasts using this formula.

n = sum(all(~isnan([X y]')));
cil = yF - norminv(0.975)*sqrt(diag(YFCov));
ciu = yF + norminv(0.975)*sqrt(diag(YFCov));

Plot the data, forecasts, and forecast intervals.

figure;
plot(dates(end-30:end),DataTable.GNPR(end-30:end));
hold on
h = gca;
plot(dates((end - fhs + 1):end),yF)
plot(dates((end - fhs + 1):end),[cil ciu],'k--')
hp = patch([dates(end - fhs + 1) dates(end) dates(end) dates(end - fhs + 1)],...
    h.YLim([1,1,2,2]),[0.8 0.8 0.8]);
uistack(hp,'bottom');
legend('Forecast horizon','True GNPR','Forecasted GNPR',...
    'Credible interval','Location','NW')
title('Real Gross National Product: 1909 - 1970');
ylabel('rGNP');
xlabel('Year');
hold off

Input Arguments

Mdl — Standard Bayesian linear regression model or model for predictor variable selection

conjugateblm model object | semiconjugateblm model object | diffuseblm model object | mixconjugateblm model object | lassoblm model object

Standard Bayesian linear regression model or model for predictor variable selection, specified as a model object in this table.

Model Object	Description
conjugateblm	Dependent, normal-inverse-gamma conjugate model returned by bayeslm or estimate
semiconjugateblm	Independent, normal-inverse-gamma semiconjugate model returned by bayeslm
diffuseblm	Diffuse prior model returned by bayeslm
empiricalblm	Prior model characterized by samples from prior distributions, returned by bayeslm or estimate
customblm	Prior distribution function that you declare returned by bayeslm
mixconjugateblm	Dependent, Gaussian-mixture-inverse-gamma conjugate model for SSVS predictor variable selection, returned by bayeslm
mixsemiconjugateblm	Independent, Gaussian-mixture-inverse-gamma semiconjugate model for SSVS predictor variable selection, returned by bayeslm
lassoblm	Bayesian lasso regression model returned by bayeslm

Typically, model objects returned by estimate represent marginal posterior distributions. When you estimate a posterior by using estimate, if you specify estimation of a conditional posterior, then estimate returns the prior model.

XF — Forecast-horizon predictor data

numeric matrix

Forecast-horizon predictor data, specified as a numPeriods-by-PriorMdl.NumPredictors numeric matrix. numPeriods determines the length of the forecast horizon. The columns of XF correspond to the columns of any other predictor data set, that is, X or data used to form the posterior distribution in Mdl.

Data Types: double

X — Predictor data

numeric matrix

Predictor data for the multiple linear regression model, specified as a numObservations-by-PriorMdl.NumPredictors numeric matrix. numObservations is the number of observations and must be equal to the length of y.

If Mdl is a posterior distribution, then the columns of X must correspond to the columns of the predictor data used to estimate the posterior.

Data Types: double

y — Response data

numeric vector

Response data for the multiple linear regression model, specified as a numeric vector with numObservations elements.

Data Types: double

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: 'Sigma2',2 specifies forecasting from the conditional predictive distribution given the specified disturbance variance of 2.

Options for All Models Except For Empirical

'Beta' — Value of regression coefficients for forecasting from conditional predictive distribution given regression coefficients

empty array ([]) (default) | numeric column vector

Value of the regression coefficients for forecasting from the conditional predictive distribution given the regression coefficients, specified as the comma-separated pair consisting of 'Beta' and an (Mdl.Intercept + Mdl.NumPredictors)-by-1 numeric vector. When using a posterior distribution, forecast forecasts from $$\pi \left(\widehat{y}|y,X,\beta =Beta\right)$$, where y is y, X is X, and Beta is the value of 'Beta'. If Mdl.Intercept is true, then Beta(1) corresponds to the model intercept. All other values correspond to the predictor variables that compose the columns of X.

You cannot specify Beta and Sigma2 simultaneously.

By default, forecast does not forecast from the conditional predictive distribution given β.

Example: 'Beta',1:3

Data Types: double

'Sigma2' — Value of disturbance variance forecasting from conditional predictive distribution given disturbance variance

empty array ([]) (default) | positive numeric scalar

Value of the disturbance variance forecasting from the conditional predictive distribution given the disturbance variance, specified as the comma-separated pair consisting of 'Sigma2' and a positive numeric scalar. When using a posterior distribution, forecast draws from $$\pi \left(\widehat{y}|y,X,{\sigma }^2=Sigma2\right)$$, where y is y, X is X, and Sigma2 is the value of 'Sigma2'.

You cannot specify Sigma2 and Beta simultaneously.

By default, forecast does not draw from the conditional predictive posterior of σ².

Example: 'Sigma2',1

Data Types: double

Options for All Models Except Conjugate

'NumDraws' — Monte Carlo simulation adjusted sample size

1e5 (default) | positive integer

Monte Carlo simulation adjusted sample size, specified as the comma-separated pair consisting of 'NumDraws' and a positive integer. forecast actually draws BurnIn – NumDraws*Thin samples. Therefore, forecast bases the estimates off NumDraws samples. For details on how forecast reduces the full Monte Carlo sample, see Algorithms.

If Mdl is a semiconjugateblm model and you specify Beta or Sigma2, then MATLAB^® ignores NumDraws.

Example: 'NumDraws',1e7

Data Types: double

Options for All Models Except Conjugate and Empirical

'BurnIn' — Number of draws to remove from beginning of Monte Carlo sample

5000 (default) | nonnegative scalar

Number of draws to remove from the beginning of the Monte Carlo sample to reduce transient effects, specified as the comma-separated pair consisting of 'BurnIn' and a nonnegative scalar. For details on how forecast reduces the full Monte Carlo sample, see Algorithms.

Tip

To help you specify the appropriate burn-in period size:

1. Determine the extent of the transient behavior in the sample by specifying 'BurnIn',0.

2. Simulate a few thousand observations by using forecast.

3. Draw trace plots.

Example: 'BurnIn',0

Data Types: double

'Thin' — Monte Carlo adjusted sample size multiplier

1 (default) | positive integer

Monte Carlo adjusted sample size multiplier, specified as the comma-separated pair consisting of 'Thin' and a positive integer.

The actual Monte Carlo sample size is BurnIn + NumDraws*Thin. After discarding the burn-in, forecast discards every Thin – 1 draws, and then retains the next draw. For details on how forecast reduces the full Monte Carlo sample, see Algorithms.

Tip

To reduce potential large serial correlation in the Monte Carlo sample, or to reduce the memory consumption of the draws stored in Mdl, specify a large value for Thin.

Example: 'Thin',5

Data Types: double

'BetaStart' — Starting values of regression coefficients for MCMC sample

numeric column vector

Starting values of the regression coefficients for the Markov chain Monte Carlo (MCMC) sample, specified as the comma-separated pair consisting of 'BetaStart' and a numeric column vector with (PriorMdl.Intercept + PriorMdl.NumPredictors) elements. By default, BetaStart is the ordinary least-squares (OLS) estimate.

Tip

A good practice is to run forecast multiple times using different parameter starting values. Verify that the solutions from each run converge to similar values.

Example: 'BetaStart',[1; 2; 3]

Data Types: double

'Sigma2Start' — Starting values of disturbance variance for MCMC sample

positive numeric scalar

Starting values of the disturbance variance for the MCMC sample, specified as the comma-separated pair consisting of 'Sigma2Start' and a positive numeric scalar. By default, Sigma2Start is the OLS residual mean squared error.

Tip

A good practice is to run forecast multiple times using different parameter starting values. Verify that the solutions from each run converge to similar values.

Example: 'Sigma2Start',4

Data Types: double

Options for Custom Models

'Reparameterize' — Reparameterization of σ² as log(σ²)

false (default) | true

Reparameterization of σ² as log(σ²) during posterior estimation and simulation, specified as the comma-separated pair consisting of 'Reparameterize' and a value in this table.

Value	Description
false	forecast does not reparameterize σ2.
true	forecast reparameterizes σ2 as log(σ2). forecast converts results back to the original scale and does not change the functional form of PriorMdl.LogPDF.

Tip

If you experience numeric instabilities during the posterior estimation or simulation of σ², then specify 'Reparameterize',true.

Example: 'Reparameterize',true

Data Types: logical

'Sampler' — MCMC sampler

'slice' (default) | 'metropolis' | 'hmc'

MCMC sampler, specified as the comma-separated pair consisting of 'Sampler' and a value in this table.

Value	Description
'slice'	Slice sampler
'metropolis'	Random walk Metropolis sampler
'hmc'	Hamiltonian Monte Carlo (HMC) sampler

Tip

• To increase the quality of the MCMC draws, tune the sampler.

  1. Before calling forecast, specify the tuning parameters and their values by using sampleroptions. For example, to specify the slice sampler width width, use:

     options = sampleroptions('Sampler',"slice",'Width',width);

  2. Specify the object containing the tuning parameter specifications returned by sampleroptions by using the 'Options' name-value pair argument. For example, to use the tuning parameter specifications in options, specify:

     'Options',options

• If you specify the HMC sampler, then a best practice is to provide the gradient for some variables, at least. forecast resorts the numerical computation of any missing partial derivatives (NaN values) in the gradient vector.

Example: 'Sampler',"hmc"

Data Types: string

'Options' — Sampler options

[] (default) | structure array

Sampler options, specified as the comma-separated pair consisting of 'Options' and a structure array returned by sampleroptions. Use 'Options' to specify the MCMC sampler and its tuning-parameter values.

Example: 'Options',sampleroptions('Sampler',"hmc")

Data Types: struct

'Width' — Typical sampling-interval width

positive numeric scalar | numeric vector of positive values

Typical sampling-interval width around the current value in the marginal distributions for the slice sampler, specified as the comma-separated pair consisting of 'Width' and a positive numeric scalar or a (PriorMdl.Intercept + PriorMdl.NumPredictors + 1)-by-1 numeric vector of positive values. The first element corresponds to the model intercept, if one exists in the model. The next PriorMdl.NumPredictors elements correspond to the coefficients of the predictor variables ordered by the predictor data columns. The last element corresponds to the model variance.

• If Width is a scalar, then forecast applies Width to all PriorMdl.NumPredictors + PriorMdl.Intercept + 1 marginal distributions.

• If Width is a numeric vector, then forecast applies the first element to the intercept (if one exists), the next PriorMdl.NumPredictors elements to the regression coefficients corresponding to the predictor variables in X, and the last element to the disturbance variance.

• If the sample size (size(X,1)) is less than 100, then Width is 10 by default.

• If the sample size is at least 100, then forecast sets Width to the vector of corresponding posterior standard deviations by default, assuming a diffuse prior model (diffuseblm).

The typical width of the slice sampler does not affect convergence of the MCMC sample. It does affect the number of required function evaluations, that is, the efficiency of the algorithm. If the width is too small, then the algorithm can implement an excessive number of function evaluations to determine the appropriate sampling width. If the width is too large, then the algorithm might have to decrease the width to an appropriate size, which requires function evaluations.

forecast sends Width to the slicesample function. For more details, see slicesample.

Tip

• For maximum flexibility, specify the slice sampler width width by using the 'Options' name-value pair argument. For example:

  'Options',sampleroptions('Sampler',"slice",'Width',width)

Example: 'Width',[100*ones(3,1);10]

Output Arguments

yF — Forecasted responses

numeric vector

Forecasted responses (the mean of the predictive distribution), returned as a numPeriods-by-1 numeric vector. Rows correspond to the rows of XF.

YFCov — Covariance matrix of predictive distribution

numeric matrix

Covariance matrix of the predictive distribution, returned as a numPeriods-by-numPeriods numeric, symmetric, positive definite matrix. Rows and columns correspond to the rows of yF.

To obtain a vector of standard deviations for the forecasted responses, enter sqrt(diag(YFCov)).

Limitations

If Mdl is an empiricalblm model object, then you cannot specify Beta or Sigma2. You cannot forecast from conditional predictive distributions by using an empirical prior distribution.

More About

Bayesian Linear Regression Model

A Bayesian linear regression model treats the parameters β and σ² in the multiple linear regression (MLR) model y_t = x_tβ + ε_t as random variables.

For times t = 1,...,T:

• y_t is the observed response.

• x_t is a 1-by-(p + 1) row vector of observed values of p predictors. To accommodate a model intercept, x_1t = 1 for all t.

• β is a (p + 1)-by-1 column vector of regression coefficients corresponding to the variables that compose the columns of x_t.

• ε_t is the random disturbance with a mean of zero and Cov(ε) = σ²I_T×T, while ε is a T-by-1 vector containing all disturbances. These assumptions imply that the data likelihood is

  $$\ell \left(\beta ,{\sigma }^2|y,x\right)={\displaystyle \prod _{t=1}^T\varphi \left(y_t;x_t\beta ,{\sigma }^2\right).}$$

  ϕ(y_t;x_tβ,σ²) is the Gaussian probability density with mean x_tβ and variance σ² evaluated at y_t;.

Before considering the data, you impose a joint prior distribution assumption on (β,σ²). In a Bayesian analysis, you update the distribution of the parameters by using information about the parameters obtained from the likelihood of the data. The result is the joint posterior distribution of (β,σ²) or the conditional posterior distributions of the parameters.

Tips

• Monte Carlo simulation is subject to variation. If forecast uses Monte Carlo simulation, then estimates and inferences might vary when you call forecast multiple times under seemingly equivalent conditions. To reproduce estimation results, before calling forecast, set a random number seed by using rng.

• If forecast issues an error while estimating the posterior distribution using a custom prior model, then try adjusting initial parameter values by using BetaStart or Sigma2Start, or try adjusting the declared log prior function, and then reconstructing the model. The error might indicate that the log of the prior distribution is –Inf at the specified initial values.

• To forecasted responses from the conditional posterior predictive distribution of analytically intractable models, except empirical models, pass your prior model object and the estimation-sample data to forecast. Then, specify the Beta name-value pair argument to forecast from the conditional posterior of σ², or specify the Sigma2 name-value pair argument to forecast from the conditional posterior of β.

Algorithms

• Whenever forecast must estimate a posterior distribution (for example, when Mdl represents a prior distribution and you supply X and y) and the posterior is analytically tractable, forecast evaluates the closed-form solutions to Bayes estimators. Otherwise, forecast resorts to Monte Carlo simulation to forecast by using the posterior predictive distribution. For more details, see Posterior Estimation and Inference.

• This figure illustrates how forecast reduces the Monte Carlo sample using the values of NumDraws, Thin, and BurnIn.

  

  Rectangles represent successive draws from the distribution. forecast removes the white rectangles from the Monte Carlo sample. The remaining NumDraws black rectangles compose the Monte Carlo sample.