expectations vs realized

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expectations vs realized

Postby KKLS » Sat Jul 15, 2017 7:53 am

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).

Mny thanks in advance
KKLS
 
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Re: expectations vs realized

Postby jpfeifer » Sat Jul 15, 2017 8:00 am

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.
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Re: expectations vs realized

Postby KKLS » Sat Jul 15, 2017 8:14 am

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.
KKLS
 
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Re: expectations vs realized

Postby jpfeifer » Sat Jul 15, 2017 8:35 am

Please provide a link to the paper.
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Re: expectations vs realized

Postby KKLS » Sat Jul 15, 2017 8:45 am

The link to the PDF:
https://www.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 .
KKLS
 
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Re: expectations vs realized

Postby jpfeifer » Sun Jul 16, 2017 9:22 am

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.
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University of Cologne
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Re: expectations vs realized

Postby KKLS » Sun Jul 16, 2017 4:16 pm

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.

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):

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
KKLS
 
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