Expectations vs realized

[KKLS]

Hi all

When writing

X(+1)

in dynare it is the expected value of X at (t+1).
I am wondering if there is a way in Dynare to distinguish between

• the expected value of X at (t+1) and
• the realized value of X at (t+1).

Many thanks in advance

[jpfeifer]

What do you have in mind? You are writing down a recursive system at time t. If X(+1) is perfectly known at time t, it is predetermined and should get a different timing in Dynare’s timing convention. Otherwise, it will be expected.

[KKLS]

Thanks for the quick response

Well, there is a (working) paper that distinguishes between E_t X(t+1) and realized X(t+1). (I can put a link of the PDF)

Obviously does that mean that in Dynare the way to understand your reply is :

• write
  X(+1)

  for E_t X(t+1) and

• write
  X

  for realized X(t+1) ?

This is what you mean when saying

If X(+1) is perfectly known at time t, it is predetermined and should get a different timing in Dynare’s timing convention.

[jpfeifer]

Please provide a link to the paper.

[KKLS]

The link to the PDF:
econstor.eu/bitstream/10419 … 918735.pdf

In page 17 of the electronic (or printed page 7 ):

he refers to a variable RL(t+1) as a function of E(t) RK(t+1) in equation (8)
he refers to the same RL(t+1) as a function of RK(t+1) in equation (9)

In eq. (9) it is state contingent because (RL) depends on realized value of RK .

So I wonder how I can make the difference n Dynare ?

Many thanks for looking at it .

[jpfeifer]

It is really a matter of timing. In equations 8 and 9, q_t, K_{t+1}^i, and L_t+1^i and omega_{t+1}^{i,a} are contained in the information set at time t, i.e. known at time t. The latter three are actually predetermined variables (loan stock, capital stock, ex-ante return). The only difference is the return to capital.

In equation 8, you have E_t(R_{t+1}^k). This expected values is known at time t as well, making the whole right-hand side known at time t. Thus, R_{t+1}^L on the left should actually get the timing R_t^L in Dynare, because it is contained in this information set.

In contrast, equation contains an R_{t+1}^k, implying the R_{t+1}^L is only contained in the information set at time t+1. But we are not trying to define an expected lending rate at time t, but the actual lending rate at time t (remember, we are defining a recursive equilibrium system to pin down variables at time t, not t+1). To make this equation state-contingent, i.e. hold for every single state realization, you have to shift the whole equation by one period to the past. You will then have an equation defining R_t^L and linking it to R_t^k and a bunch of predetermined variables.

[KKLS]

This post is edited!

1. You say :

It is really a matter of timing. In equations 8 and 9, q_t, K_{t+1}^i, and L_t+1^i and omega_{t+1}^{i,a} are contained in the information set at time t, i.e. known at time t. The latter three are actually predetermined variables (loan stock, capital stock, ex-ante return). The only difference is the return to capital.

In equation 8, you have E_t(R_{t+1}^k). This expected values is known at time t as well, making the whole right-hand side known at time t. Thus, R_{t+1}^L on the left should actually get the timing R_t^L in Dynare, because it is contained in this information set.

Let me assume that I do not use the predetermined_variables command (hence K_t = (1-delta) K_(t-1) + INV_t )… then all the terms in equation (8) will be written with subscript (t), except for ‘’ Rk_t+1 ‘’ which will take the subscript ‘‘t+1’’ in Dynare ? Correct ??

RL_t = Rk_t+1 * omega_t * q_t *K_t *L_t (eq. 8 )

2. Question: Then in the last paragraph:

In contrast, equation (you meant 9) contains an R_{t+1}^k, implying the R_{t+1}^L is only contained in the information set at time t+1. But we are not trying to define an expected lending rate at time t, but the actual lending rate at time t (remember, we are defining a recursive equilibrium system to pin down variables at time t, not t+1). To make this equation state-contingent, i.e. hold for every single state realization, you have to shift the whole equation by one period to the past. You will then have an equation defining R_t^L and linking it to R_t^k and a bunch of predetermined variables.[/quote]

The way I understand this is (assuming I do not use predetermined_variables command), I should write in Dynare :

RL_t = Rk_t * omega_t * q_t-1 *K_t-1 *L_t-1 (eq. 9)

Is this reading correct ?

If it is then I will not be able to derive equation (12) in the working paper (in page 8 in print, or page 18 in electronic numbering)

Question 3.
A final confirmation. You say ''omega_{t+1}^{i,a} ‘’ is predetermined. the author of the working paper says in page 5 (printed page 5, electronic page15):

[quote] where expectations are taken with respect to the random variable Rk_t+1, and ω^i_t+1 is a function of realization of Rk_t+1 (and therefore, function of the states).

Is he trying to say exactly this: that ''omega_{t+1}^{i,a} ‘’ is predetermined ? Unlike in BGG original paper who take expectations w.r.t both, ''omega_{t+1}^{i,a} ‘’ and ‘‘Rk_t+1’’.

Am I reading it right ?

Regards