ecmmvnrstd

Evaluate standard errors for multivariate normal regression model

Syntax

[StdParameters,StdCovariance] = ecmmvnrstd(Data,Design,Covariance)

[StdParameters,StdCovariance] = ecmmvnrstd(___,Method,CovarFormat)

Description

[StdParameters,StdCovariance] = ecmmvnrstd(Data,Design,Covariance) evaluates standard errors for a multivariate normal regression model with missing data. The model has the form

$$Dat{a}_k\sim N\left(Desig{n}_k\times Parameters,Covariance\right)$$

for samples k = 1, ... , NUMSAMPLES.

[StdParameters,StdCovariance] = ecmmvnrstd(___,Method,CovarFormat) adds an optional arguments for Method and CovarFormat.

Examples

Compute Standard Errors for Multivariate Normal Regression

This example shows how to compute standard errors for a multivariate normal regression model.

First, load dates, total returns, and ticker symbols for the twelve stocks from the MAT-file.

load CAPMuniverse
whos Assets Data Dates

  Name           Size             Bytes  Class     Attributes

  Assets         1x14              1792  cell                
  Data        1471x14            164752  double              
  Dates       1471x1              11768  double              

Dates = datetime(Dates,'ConvertFrom','datenum');

The assets in the model have the following symbols, where the last two series are proxies for the market and the riskless asset.

Assets(1:14)

ans = 1×14 cell
    {'AAPL'}    {'AMZN'}    {'CSCO'}    {'DELL'}    {'EBAY'}    {'GOOG'}    {'HPQ'}    {'IBM'}    {'INTC'}    {'MSFT'}    {'ORCL'}    {'YHOO'}    {'MARKET'}    {'CASH'}

The data covers the period from January 1, 2000 to November 7, 2005 with daily total returns. Two stocks in this universe have missing values that are represented by NaNs. One of the two stocks had an IPO during this period and, consequently, has significantly less data than the other stocks.

[Mean,Covariance] = ecmnmle(Data);

Compute separate regressions for each stock, where the stocks with missing data have estimates that reflect their reduced observability.

[NumSamples, NumSeries] = size(Data);
NumAssets = NumSeries - 2;

StartDate = Dates(1);
EndDate = Dates(end);

Alpha = NaN(1, length(NumAssets));
Beta = NaN(1, length(NumAssets));
Sigma = NaN(1, length(NumAssets));
StdAlpha = NaN(1, length(NumAssets));
StdBeta = NaN(1, length(NumAssets));
StdSigma = NaN(1, length(NumAssets));
for i = 1:NumAssets
	% Set up separate asset data and design matrices
	TestData = zeros(NumSamples,1);
	TestDesign = zeros(NumSamples,2);

	TestData(:) = Data(:,i) - Data(:,14);
	TestDesign(:,1) = 1.0;
	TestDesign(:,2) = Data(:,13) - Data(:,14);

    [Param, Covar] = ecmmvnrmle(TestData, TestDesign);
    
	% Estimate the sample standard errors for model parameters for each asset.
	StdParam = ecmmvnrstd(TestData, TestDesign, Covar,'hessian')
    
end   

StdParam = 2×1

    0.0008
    0.0715

StdParam = 2×1

    0.0012
    0.1000

StdParam = 2×1

    0.0008
    0.0663

StdParam = 2×1

    0.0007
    0.0567

StdParam = 2×1

    0.0010
    0.0836

StdParam = 2×1

    0.0014
    0.2159

StdParam = 2×1

    0.0007
    0.0567

StdParam = 2×1

    0.0004
    0.0376

StdParam = 2×1

    0.0007
    0.0585

StdParam = 2×1

    0.0005
    0.0429

StdParam = 2×1

    0.0008
    0.0709

StdParam = 2×1

    0.0010
    0.0853

Input Arguments

Data — Data

matrix

Data, specified as an NUMSAMPLES-by-NUMSERIES matrix with NUMSAMPLES samples of a NUMSERIES-dimensional random vector. Missing values are indicated by NaNs. Only samples that are entirely NaNs are ignored. (To ignore samples with at least one NaN, use mvnrmle.)

Data Types: double

Design — Design model

matrix | cell array

Design model, specified as a matrix or a cell array that handles two model structures:

• If NUMSERIES = 1, Design is a NUMSAMPLES-by-NUMPARAMS matrix with known values. This structure is the standard form for regression on a single series.

• If NUMSERIES ≥ 1, Design is a cell array. The cell array contains either one or NUMSAMPLES cells. Each cell contains a NUMSERIES-by-NUMPARAMS matrix of known values.

  If Design has a single cell, it is assumed to have the same Design matrix for each sample. If Design has more than one cell, each cell contains a Design matrix for each sample.

Data Types: double | cell

Covariance — Estimates for covariance of regression residuals

matrix

Estimates for the covariance of the regression residuals, specified as an NUMSERIES-by-NUMSERIES matrix.

Data Types: double

Method — Method of calculation for the information matrix

'hessian' (default) | character vector

(Optional) Method of calculation for the information matrix, specified as a character vector defined as:

• 'hessian' — The expected Hessian matrix of the observed log-likelihood function. This method is recommended since the resultant standard errors incorporate the increased uncertainties due to missing data.

• 'fisher' — The Fisher information matrix.

  Note

  If Method = 'fisher', to obtain more quickly just the standard errors of variance estimates without the standard errors of the covariance estimates, set CovarFormat = 'diagonal' regardless of the form of the covariance matrix.

Data Types: char

CovarFormat — Format for the covariance matrix

'full' (default) | character vector

(Optional) Format for the covariance matrix, specified as a character vector. The choices are:

• 'full' — Compute the full covariance matrix.

• 'diagonal' — Force the covariance matrix to be a diagonal matrix.

Data Types: char

Output Arguments

StdParameters — Standard errors for each element of Parameters

vector

Standard errors for each element of Parameters, returned as an NUMPARAMS-by-1 column vector.

StdCovariance — Standard errors for each element of Covariance

matrix

Standard errors for each element of Covariance, returned as an NUMSERIES-by-NUMSERIES matrix.

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.

References

[1] Little, Roderick J. A. and Donald B. Rubin. Statistical Analysis with Missing Data. 2nd Edition. John Wiley & Sons, Inc., 2002.

Version History

Introduced in R2006a