convert2sur

Convert multivariate normal regression model to seemingly unrelated regression (SUR) model

Syntax

DesignSUR = convert2sur(Design,Group)

Description

DesignSUR = convert2sur(Design,Group) converts a multivariate normal regression model into a seemingly unrelated regression model with a specified grouping of the data series.

Examples

Use convert2sur to Estimate Stock Alpha and Beta Values

This example shows a CAPM demonstration using 6 stocks and 60 months of simulated asset returns, where the model for each stock is AssetReturn = Alpha * 1 + CashReturn + Beta * (MarketReturn - CashReturn) + Noise and the parameters to estimate are Alpha and Beta.

Using simulated data, where the Alpha estimate(s) are displayed in the first row(s) and the Beta estimate(s) are display in the second row(s).

Market = (0.1 - 0.04) + 0.17*randn(60, 1);
Asset = (0.1 - 0.04) + 0.35*randn(60, 6);

Design = cell(60, 1);
for i = 1:60
            Design{i} = repmat([ 1, Market(i) ], 6, 1);
end

Obtain the aggregate estimates for all stocks.

[Param, Covar] = mvnrmle(Asset, Design);

disp({'All 6 Assets Combined'});

    {'All 6 Assets Combined'}

disp(Param);

    0.0233
    0.1050

Estimate parameters for individual stocks using convert2sur

Group = 1:6;
DesignSUR = convert2sur(Design, Group);
[Param, Covar] = mvnrmle(Asset, DesignSUR);
Param = reshape(Param, 2, 6);

disp({ 'A', 'B', 'C', 'D', 'E', 'F' });

    {'A'}    {'B'}    {'C'}    {'D'}    {'E'}    {'F'}

disp(Param);

    0.0144    0.0270    0.0046    0.0419    0.0376    0.0291
    0.3264   -0.1716    0.3248   -0.0630   -0.0001    0.0637

Estimate parameters for pairs of stocks by forming groups.

disp({'A & B', 'C & D','E & F'});

    {'A & B'}    {'C & D'}    {'E & F'}

Group = { [1,2 ],[3,4],[5,6]};
DesignSUR = convert2sur(Design, Group);
[Param, Covar] = mvnrmle(Asset, DesignSUR);

Param = reshape(Param, 2, 3);

disp(Param);

    0.0186    0.0190    0.0334
    0.0988    0.1757    0.0293

Input Arguments

Design — Data series

matrix | cell array

Data series, specified as a matrix or a cell array that depends on the number of data series NUMSERIES.

• If NUMSERIES = 1, convert2sur returns the Design matrix.

• If NUMSERIES > 1, Design is a cell array with NUMSAMPLES cells, where each cell contains a NUMSERIES-by-NUMPARAMS matrix of known values.

Data Types: double | cell

Group — Grouping for data series

vector | cell array

Grouping for data series, specified using separate parameters for each group. Specify groups either by series or by groups:

• To identify groups by series, construct an index vector that has NUMSERIES elements. Element i = 1, ..., NUMSERIES in the vector, and has the index j = 1, ..., NUMGROUPS of the group in which series i is a member.

• To identify groups by groups, construct a cell array with NUMGROUPS elements. Each cell contains a vector with the indexes of the series that populate a given group.

  In either case, the number of series is NUMSERIES and the number of groups is NUMGROUPS, with 1 ≤ NUMGROUPS ≤ NUMSERIES.

Data Types: double | cell

Output Arguments

DesignSUR — Seemingly unrelated regression model with a specified grouping of the data series

matrix | cell array

Seemingly unrelated regression model with a specified grouping of the data series, returned as either a matrix or a cell array that depends on the value of NUMSERIES.

• If NUMSERIES = 1, DesignSUR = Design, which is a NUMSAMPLES-by-NUMPARAMS matrix.

• If NUMSERIES > 1 and NUMGROUPS groups are to be formed, Design is a cell array with NUMSAMPLES cells, where each cell contains a NUMSERIES-by-(NUMGROUPS * NUMPARAMS) matrix of known values.

The original collection of parameters that are common to all series are replicated to form collections of parameters for each group.

More About

Multivariate Normal Regression

Multivariate normal regression is a statistical modeling approach that extends traditional regression techniques to situations where the dependent variable is multivariate, meaning it consists of multiple correlated outcomes.

Multivariate normal regression is particularly useful in fields where researchers are interested in understanding the relationships between several response variables and one or more predictor variables.

Seemingly Unrelated Regression Model

The Seemingly Unrelated Regression (SUR) model is a statistical method used in econometrics to estimate multiple regression equations that are related to each other through their error terms.

The SUR model allows for the simultaneous estimation of these equations, taking into account the potential correlation between the error terms of different equations. This can lead to more efficient estimates compared to estimating each equation independently.

Version History

Introduced in R2006a