g13dn:: Time Series Analysis (NAG Toolbox)

Let

W_{t} = {(w_{1 t}, w_{2 t}, \dots, w_{k t})}^{T}

, for

t = 1, 2, \dots, n

, denote

n

observations of a vector of

k

time series. The partial lag correlation matrix at lag

l

P (l)

, is defined to be the correlation matrix between

W_{t}

and

W_{t + l}

, after removing the linear dependence on each of the intervening vectors

W_{t + 1}, W_{t + 2}, \dots, W_{t + l - 1}

. It is the correlation matrix between the residual vectors resulting from the regression of

W_{t + l}

on the carriers

W_{t + l - 1}, \dots, W_{t + 1}

and the regression of

W_{t}

on the same set of carriers; see Heyse and Wei (1985).

P (l)

has the following properties.

(i)	If $W_{t}$ follows a vector autoregressive model of order $p$ , then $P (l) = 0$ for $l > p$ ;
(ii)	When $k = 1$ , $P (l)$ reduces to the univariate partial autocorrelation at lag $l$ ;
(iii)	Each element of $P (l)$ is a properly normalized correlation coefficient;
(iv)	When $l = 1$ , $P (l)$ is equal to the cross-correlation matrix at lag $1$ (a natural property which also holds for the univariate partial autocorrelation function).

Sample estimates of the partial lag correlation matrices may be obtained using the recursive algorithm described in Wei (1990). They are calculated up to lag

m

, which is usually taken to be at most

n / 4

. Only the sample cross-correlation matrices (

\hat{R} (l)

, for

l = 0, 1, \dots, m

) and the standard deviations of the series are required as input to nag_tsa_multi_corrmat_partlag (g13dn). These may be computed by nag_tsa_multi_corrmat_cross (g13dm). Under the hypothesis that

W_{t}

follows an autoregressive model of order

s - 1

, the elements of the sample partial lag matrix

\hat{P} (s)

, denoted by

{\hat{P}}_{i j} (s)

, are asymptotically Normally distributed with mean zero and variance

1 / n

. In addition the statistic

X (s) = n \sum_{i = 1}^{k} \sum_{j = 1}^{k} {\hat{P}}_{i j} {(s)}^{2}

has an asymptotic

χ^{2}

-distribution with

k^{2}

degrees of freedom. These quantities,

X (l)

, are useful as a diagnostic aid for determining whether the series follows an autoregressive model and, if so, of what order.

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

Accuracy

Further Comments

The function nag_tsa_multi_autocorr_part (g13db) computes squared partial autocorrelations for a specified number of lags. It may also be used to estimate a sequence of partial autoregression matrices at lags

1, 2, \dots

by making repeated calls to the function with the argument nk set to

1, 2, \dots

. The

(i, j)

th element of the sample partial autoregression matrix at lag

l

is given by

W (i, j, l)

when nk is set equal to

l

on entry to nag_tsa_multi_autocorr_part (g13db). Note that this is the ‘Yule–Walker’ estimate. Unlike the partial lag correlation matrices computed by nag_tsa_multi_corrmat_partlag (g13dn), when

W_{t}

follows an autoregressive model of order

s - 1

, the elements of the sample partial autoregressive matrix at lag

s

do not have variance

1 / n

, making it very difficult to spot a possible cut-off point. The differences between these matrices are discussed further by Wei (1990).

Example

This example computes the sample partial lag correlation matrices of two time series of length

48

, up to lag

10

. The matrices, their

χ^{2}

-statistics and significance levels and a plot of symbols indicating which elements of the sample partial lag correlation matrices are significant are printed. Three * represent significance at the

0.5

% level, two * represent significance at the 1% level and a single * represents significance at the 5% level. The * are plotted above or below the central line depending on whether the elements are significant in a positive or negative direction.

function g13dn_example


fprintf('g13dn example results\n\n');

w = [-1.49, -1.62, 5.20, 6.23, 6.21, 5.86, 4.09, 3.18, 2.62, 1.49, 1.17, ...
      0.85, -0.35, 0.24, 2.44, 2.58, 2.04, 0.40, 2.26, 3.34, 5.09, 5.00, ...
      4.78,  4.11, 3.45, 1.65, 1.29, 4.09, 6.32, 7.50, 3.89, 1.58, 5.21, ...
      5.25,  4.93, 7.38, 5.87, 5.81, 9.68, 9.07, 7.29, 7.84, 7.55, 7.32, ...
      7.97,  7.76, 7.00, 8.35;
      7.34,  6.35, 6.96, 8.54, 6.62, 4.97, 4.55, 4.81, 4.75, 4.76,10.88, ...
     10.01, 11.62,10.36, 6.40, 6.24, 7.93, 4.04, 3.73, 5.60, 5.35, 6.81, ...
      8.27,  7.68, 6.65, 6.08,10.25, 9.14,17.75,13.30, 9.63, 6.80, 4.08, ...
      5.06,  4.94, 6.65, 7.94,10.76,11.89, 5.85, 9.01, 7.50,10.02,10.38, ...
      8.15, 8.37, 10.73, 12.14];
[k,n] = size(w);
k = int64(k);
n = int64(n);
m = int64(10);
matrix = 'R';

% Calculate cross correlations
[wmean, r0, r, ifail] = g13dm( ...
			       matrix, k, m, w);

% Calculate sample partial lag correlation matrices
[maxlag, parlag, x, pvalue, ifail] = ...
  g13dn( ...
	 n, m, r0, r);

disp('Partial Lag Correlation Matrices');
for l = 1:m
  fprintf('Lag = %d\n',l);
  disp(parlag(:,:,l));
end
sn1 = 1/sqrt(double(n));
fprintf('Standard error = 1/sqrt(n) = %7.4f\n\n',sn1);

disp('Tables Of Indicator Symbols');
fprintf('\nFor Lags 1 to %d\n',m);
lhs = {'              0.005  :'; '        +     0.01   :';
       '              0.05   :'; 
       '   Sig. Level        :- - - - - - - - - -  Lags';
       '              0.05   :';
       '        -     0.01   :'; '              0.005  :'};
c = sn1*[3.29, 2.58, 1.96, 0, -1.96, -2.58, -3.29];
for i = 1:k
  for j=1:k
    if i==j
      fprintf('\nAuto-correlation function for series %d\n', i);
    else
      fprintf('\nCross-correlation function for series %d and series %d\n', ...
	      i, j);
    end
    rhs = lhs;
    for t = 1:m
      for u = 1:3
	if parlag(i,j,t)>c(u)
	  rhs{u} = strcat(rhs{u},'*');
	end
      end
      for u = 5:7
	if parlag(i,j,t)<c(u)
	  rhs{u} = strcat(rhs{u},'*');
	end
      end
    end
    fprintf('\n');
    fprintf('%s\n',rhs{1:end});
  end
end  

fprintf('\n Lag     Chi-square statistic     P-value\n\n');
ilag = double([1:m]);
fprintf('%4d%18.3f%19.4f\n',[ilag; x'; pvalue']);

g13dn example results

Partial Lag Correlation Matrices
Lag = 1
    0.7359    0.1743
    0.2114    0.5546

Lag = 2
   -0.1869   -0.0832
   -0.1805   -0.0724

Lag = 3
    0.2775   -0.0069
    0.0837   -0.2133

Lag = 4
   -0.0843    0.2269
    0.1284   -0.1764

Lag = 5
    0.2361    0.2384
   -0.0468   -0.0455

Lag = 6
   -0.0164    0.0873
    0.0996   -0.0809

Lag = 7
   -0.0355    0.2611
    0.1258    0.0120

Lag = 8
    0.0767    0.3814
    0.0268   -0.1492

Lag = 9
   -0.0651   -0.3868
    0.1887    0.0564

Lag = 10
   -0.0261   -0.2861
    0.0279   -0.1729

Standard error = 1/sqrt(n) =  0.1443

Tables Of Indicator Symbols

For Lags 1 to 10

Auto-correlation function for series 1

              0.005  :*
        +     0.01   :*
              0.05   :*
   Sig. Level        :- - - - - - - - - -  Lags
              0.05   :
        -     0.01   :
              0.005  :

Cross-correlation function for series 1 and series 2

              0.005  :
        +     0.01   :*
              0.05   :*
   Sig. Level        :- - - - - - - - - -  Lags
              0.05   :**
        -     0.01   :*
              0.005  :

Cross-correlation function for series 2 and series 1

              0.005  :
        +     0.01   :
              0.05   :
   Sig. Level        :- - - - - - - - - -  Lags
              0.05   :
        -     0.01   :
              0.005  :

Auto-correlation function for series 2

              0.005  :*
        +     0.01   :*
              0.05   :*
   Sig. Level        :- - - - - - - - - -  Lags
              0.05   :
        -     0.01   :
              0.005  :

 Lag     Chi-square statistic     P-value

   1            44.363             0.0000
   2             3.825             0.4302
   3             6.220             0.1833
   4             5.096             0.2776
   5             5.609             0.2303
   6             1.169             0.8832
   7             4.098             0.3929
   8             8.368             0.0790
   9             9.248             0.0552
  10             5.434             0.2456

On entry,	$k < 1$ ,
or	$n < 2$ ,
or	$m < 1$ ,
or	$m \geq n$ ,
or	$kmax < k$ ,
or	$lwork < (5 m + 6) k^{2} + k$ .

NAG Toolbox: nag_tsa_multi_corrmat_partlag (g13dn)

▸▿ Contents

Purpose

Syntax

Description