Question

plz provide two example of Econometric issues with simultaneous systems

Answer 1

1.

Consider situation of an ideal market where transaction of only one commodity, say wheat, takes place.

Assume that the number of buyers and sellers is large so that the market is a perfectly competitive market. It
is also assumed that the amount of wheat that comes into the market in a day is completely sold out on the
same day. No seller takes it back. Now we develop a model for such mechanism.
Let
d_t denotes the demand of the commodity, say wheat, at time t,
s_{​​​​​​t} denotes the supply of the commodity, say wheat, at time t, and
q_{​​​​​​t} denotes the quantity of the commodity, say wheat, transacted at time t.
By economic theory about the ideal market, we have the following condition:
,d_{​​​​​​t} =s_{​​​​​​t} .t=1............n.

Observe that
 the demand of wheat depends on
- price of wheat (p_{​​​​​​t} ) at time . t
- income of buyer (i_{​​​​​​t} ) at time . t

 the supply of wheat depends on
- price of wheat (p_{​​​​​t} ) at time t.
- rainfall (r_{​​​​​​t} ) at time t .
From market conditions, we have
. q_{​​​​​​t} =d_{​​​​​​t}= s_{​​​​​​t.}
Demand, supply and price are determined from each other.
Note that
income can influence demand and supply, but demand and supply cannot influence the income.
supply is influenced by rainfall, but rainfall is not influenced by the supply of wheat.
Our aim is to study the behaviour of s_{​​​​​​t} , p_{​​​​​​t} and r_{​​​​​​t} which are determined by the simultaneous equation
model.
Since endogenous variables are influenced by exogenous variables but not vice versa, so s_{​​​​​​t,}p_{​​​​​​t}
and r_{​​​​​​t} are endogenous variables.
i_t and r_{​​​​​​t} are exogenous variables.
Now consider an additional variable for the model as lagged value of price p_{​​​​​​t}, denoted as p_{​​​​​​t-1}. In a market,
generally the price of the commodity depends on the price of the commodity on previous day. If the price of
commodity today is less than the previous day, then buyer would like to buy more. For seller also, today’s
price of commodity depends on previous day’s price and based on which he decides the quantity of
commodity (wheat) to be brought in the market.

2.

The general linear simultaneous equations model with m equations can be written
formally as
Byt + Γzt = ut
, t = 1, · · · , T (2.1)
where yt
is an m×1 vector of observations on the m current endogenous variables
at period t, zt
is a q×1 vector of observations on the q predetermined variables, ut
is an m×1 vector of disturbances, B is a m×m square matrix of coefficients on the
endogenous variables and Γ is an m×q matrix of coefficients on the predetermined
variables.

Consider the two equation dynamic demand–supply model . In each equation,
a single variable is excluded (Wt
in the demand equation and Yt
in the second).

Thus both equations satisfy the order condition for identification.
Consider now the rank condition. for the first equation, the matrix to be
considered is the 1 × 1 matrix
[−β₃] .
Clearly this will have rank m − 1 = 1 as long as β3 6= 0. If β_{​​​​3} = 0 however, then
the equation will not be identified. Similarly, the rank condition for the second
equation is that the 1 × 1 matrix
[−α₃]
has rank m − 1 = 1. Again, this will be the case except if α_{​​​​​​3} = 0.