Common Shocks and something else

[salkiewi]

Hi there,

I have a question concerning the 'shocks' block. That is, I have the following dynamics:

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        da(+1) = mu + x + zeta*w + exp(v)*sigma_a*e_a(+1);
        x(+1)  = rho_x* x + exp(v)*sigma_x*e_x(+1);
        v      = rho_v*v(-1) - sigma_v*e_x;
        dw(+1) = mu + (rho_w - 1) * (w - a) + kappa*x + sigma_w*e_w(+1);

in the model part. I also have the values for the sigmas, which I declared in the parameters block

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                sigma_a        = 0.0052;
                sigma_v        = 0.01;
                sigma_w        = 0.024;
                sigma_x        = 0.00052

And: e_a and e_x are correlated: rho_a,x = 0.3

My questions:

1) How do I proceed in the shock part if I have the sigmas?

a) Do I just write

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var e_a = sigma_a^2;           // Volatility of Short Run Shocks 
var e_x = sigma_x^2;          // Volatility of Long Run Shocks
corr e_a, e_x = 0.3;         // Correlation of Short and Long Run Shocks
var e_w = sigma_w^2;            // Volatility of Oil Supply;

b) And if I do so, do I have to omit the sigma-terms in the dynamics? e.g. omit 'sigma_a*e_a(+1)' in da(+1)?
Or how do I do it?

2)My suggestion is probably partly wrong since - as one can see - v and x(+1) share a common shock, i.e. e_x.
Is it possible to maybe do it like this?

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var e_a(+1) = sigma_a^2;           
var e_x(+1) = sigma_x^2;         
corr e_a(+1), e_x(+1) = 0.3;
var e_x = sigma_v^2;
var e_w(+1) = sigma_w^2;


Would be really great if someone could help me.
I would highly appreciate it.

Best,
Stefan

[jpfeifer]

You can implement a correlated shock by adding a common shock, but I would go for the regular interface. For this purpose, use

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shocks;
var e_a = sigma_a^2;           // Volatility of Short Run Shocks
var e_x = sigma_x^2;          // Volatility of Long Run Shocks
corr e_a, e_x = 0.3;         // Correlation of Short and Long Run Shocks
var e_w = sigma_w^2;            // Volatility of Oil Supply;
end;

This specifies the covariance matrix of your shock. Because of this, do not put sigmas in the model block as you variance would then become sigma_a^4.

Lastly, your timing is weird. It almost never happens that you have leaded shocks enter equations. In the shocks-block, this should not even be allowed by Dynare. It looks as if you try to specify exogenous processes. They must be entered in the form

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x=rho*x(-1)+epsilon

and not

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x(+1)=rho*x+epsilon(+1)

[salkiewi]

Great! Many thanks for your helpful response!

Just one tiny little request:
What do I do with the sigma_v from the volatility process, v?
Do I just leave this volatility in the model block but omit the others if I specify the shock block as you suggested?

Many many thanks in advance!

[salkiewi]

I just encountered new difficulties.
That is, I have some problems to figure out how to correctly code the '∆w' process.

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        da     = mu + x(-1)  + zeta*w + exp(v)*e_a;
        x      = rho_x * x(-1) + exp(v)*e_x;
        v      = rho_v*v(-1) - sigma_v*e_x;
        dw     = mu + (rho_w - 1) * (w(-1) - a(-1)) + kappa*x(-1) + e_w;


The problem is that within the process, I have 'a(-1)' and 'w(-1)' as log levels rather than growth rates.
My guess is that Dynare will not like it when I try to run the code since these variables are nowhere specified, right?
How could solve this? Just by writing:

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a(-1)  = a - da
w(-1) = w -dw


But wouldnt that just be the same problem?; just rephrased since I dont have a and w after all?

Btw,

da = Productivity growth process (labor augmenting)
x = Long run risk component
v = Stochastic vola
dw= Oil supply growth process

In the model, I just have the identity

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W = G + O


--> Oil supply is allocated to the household sector for 'oil consumption' (G_t) and to the firm sector in which it enters the production function (O_t).

Many many thanks for any comments and suggestions!
Would highly appreciate it

[jpfeifer]

Regarding the sigma_v, it does not matter what you do. Either you set

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 var e_v = sigma_v^2;


in the shocks block and delete the sigma_v in the model or you set

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 var e_v = 1;


and keep the sigma_v. What you cannot do is leave sigma_v from the shocks block as this will set the variance to 0.
Regarding the second question: you need to find recursive representations or definitions for all variables in your model. If da and dw are exogenous processes, you can use the random walk definitions you posted.

[salkiewi]

Many thanks for the response. Helps a lot!
Just a little request, again. (Sorry for that.)

--> What I don't get is the following:

Let's say I incorporate sigma_v as you mentioned:

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 var e_v = sigma_v^2;


From my point of view, there is the problem that I dont have "e_v" in my dynamics - which, btw, are exogenous.
Or doesn't it matter?
Sorry, if this might seem like a lame question. But I don't get how it would affect v if I do it like this since v is exposed to e_x.

Many thanks in advance.


Ps.: Concerning the second question. Perfect. I gonna try it like this.

[jpfeifer]

I may have misunderstood you. e_x seems to be a shock that jointly enters two equations. Of course, you can only specify one variance for it in the shocks block. The sigma_v in the law of motion for v only rescales the standard deviation. Thus, you want

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model;
...
x  = rho_x* x(-1) + exp(v)*e_x;
v  = rho_v*v(-1) - sigma_v/sigma_x*e_x;
end;

shocks;
var e_a = sigma_a^2;           // Volatility of Short Run Shocks
var e_x = sigma_x^2;          // Volatility of Long Run Shocks
corr e_a, e_x = 0.3;         // Correlation of Short and Long Run Shocks
var e_w = sigma_w^2;            // Volatility of Oil Supply;
end;


That way, you have the correlation between e_a and e_x and that the shock to e_x enters both equations. At the same time, the shock term e_x in the first equation has variance sigma_x^2, while in the second equation it is
var(sigma_v/sigma_x*e_x)=sigma_v^2/sigma_x^2*sigma_x^2=sigma_v^2.

[salkiewi]

Great! Many many thanks!