III. Methodology
In the present paper, we use the same variables and sample period during the U.S. Great Depression as Bernanke. The variables are, namely, an index of industrial production IN,
wholesale price index PX, nominal money stock MS, liabilities of failed businesses LB, and the deposits of failed banks DB. The index of industrial production and money supply
M1 are seasonally adjusted, and the liabilities of failed businesses and deposits of failed banks are deflated by the wholesale price index. A logarithmic transformation of all variables
is used. The sample data is monthly and covers the period January 1921 to December 1941.
It should be noted that the deposits of failed banks do not measure lost deposits to the extent that deposits of failed banks are shifted to other banks. Recent evidence by Mason
1998 indicates that payouts to depositors of failed banks during the Great Depression were only about 30– 40 in the year of failure. In subsequent years, recoveries due to
restorations to solvency and assisted mergers i.e., purchase and assumption resolutions yielded eventual payouts of about 80 of deposits over five or more years. Interestingly,
Mason 1998 found that these statistics are similar in level and time pattern to the bank failure episode of the late 1980s and early 1990s. Comparing the two periods, he observed
that the disposition of bank assets in the resolution process can cause significant macro- economic losses. We infer that bank failures seriously disrupted the credit intermediation
process for an extended period of time during the Great Depression. For our purposes, as observed by an anonymous referee, it should be recognized that the deposits of failed
banks is a proxy for unobservable lost deposits, which are assumed to be a constant percentage over time. Also, consistent with the arguments of Bernanke 1983, this
variable is a proxy for the degree of disruption in the bank credit intermediation process.
Unlike the single equation method used in Bernanke 1983, the VAR methodology employed here generates the historical decomposition of the variables under investigation,
which shows the monthly estimates of each variable’s influence on all other variables. Thus, VAR provides additional detail on the economic and financial interactions during
the Great Depression.
Assume that the dynamic behavior of vector Y
t
5 [IN, PX, MS, LB, DB] is given by the following structural VAR model:
Y
t
5
O
k5 n
A
k
Y
t2k
1 e
t
, 1
where A , . . . , A
k
are matrices of coefficients to be estimated, and Ee
t
e9
t
5 V is diagonal. The reduced form associated with equation 1 is given by:
Y
t
5
O
k5 1
n
B
k
Y
t2k
1 u
t
, 2
where B
k
5 [I 2 A
o 21
A
i
], and V 5 [I 2 A
o 21
] ¥ [I 2 A
o 21
]9 and the vector, u
t
, represents the unexpected movements in variables in vector Y. In reduced form, VAR has
no contemporaneous term, such that each equation is estimated by ordinary least squares OLS method.
From a VAR system specified in equation 2, we can estimate the historical decom- positions of the series under investigation, which are defined as the effects of innovations
or shocks in one variable on each of the other variables in the system. These decom-
220
A. Anari and J. Kolari
positions are estimated based on the following partition of the moving average represen- tation of the variables in the system:
Y
T1J
5
O
s 5 j 2
1
A
s
u
T1j2s
1 X
T1j
b 1
O
s 5 j
`
A
s
u
T1j2s
, 3
where ¥
s5 j2
1
A
s
u
T1j2s
represents that part of Y
T1J
due to innovations in period T 1 1 to T 1 j
, and the term [ z ] is the forecast of Y
T1J
based on information available at time T. X
T1j
is the deterministic part of the model e.g., a constant or time trend term. We should comment that the forecasts are in-sample as opposed to out-of-sample and, therefore, are
intended to provide insights into how shocks in one variable e.g., a sharp decline in the deposits of failed banks affected other variables e.g., economic output.
Prior to estimating the vector autoregressions, we employed unit root tests to determine whether the data were stationary, as well as the Johansen trace test for cointegration.
Cointegration tests help to determine whether the VAR model must be estimated in level or first difference, or structured in the class of vector error-correction VEC models. The
Johansen 1991 procedure starts with the kth order unrestricted VAR model in equation 2 and transforms it to the VEC representation:
DY
t
5 m 1
O
i k2
1
G
i
DY
t2i
1
P
Y
t2k
1 e
t
, 4
where G
i
5 2 I 1 p
t
1 p
2
1 . . . 1 p
i
;
P
5 2 I 1 p
t
1 p
2
1 . . . 1 p
k
for I 5 1, . . . , k 2 1; m 5 n 3
1 vector of intercepts; e
t
5 P Np~
0, s. The difference between the VAR model in equation 2 and the VEC model in equation
4 is the term P. The coefficient matrix, P, conveys information about long-term relationships among the variables and lends itself to hypothesis testing. There are three
possible cases: 1 if matrix P is of full rank, Y is a stationary system i.e., all elements of Y
are stationary in levels; 2 if the matrix P has rank zero, all variables in Y have unit roots and equation 4 is simply a difference vector time series model; and 3 in the
intermediate case, when 0 , rankP 5 r , n, there are r cointegrating relations among the variables. In case 3, rank r , n implies that P 5 ab9, where a and b are n 3 r
matrices, and r is the number of cointegrating relationships [see Johansen 1991]. The b matrix is made up of r cointegration vectors that satisfy equation 4, and the a matrix
contains error correction parameters.
The cointegration restrictions are satisfied asymptotically by levels VAR [see Engle and Granger 1987]. Thus, we estimated a VEC model and fit a levels VAR to the same
data and chose between VEC or VAR, based on the Akaike 1973 and Schwarz 1978 criteria. If the results for VAR are similar to VEC, then there is more confidence in the
robustness of the estimated model.
Bank Failures and Economic Output
221
IV. Empirical Results
Kategori