5
votes
Accepted
Stochastic AK model derivation
I think you may substitute it directly as mentioned?
$$v(A,k) = \max log(c) + \beta E[v(A',k')|A]$$
Applying the value function expression (note period change):
$v(A',k') = \frac{log(k')}{1-\beta} + v(...
- 517
3
votes
Accepted
Bayes’ rule in "The sources of capital misallocation"
It follows immediately from Bayesian Updating using Bayes Rule for normal random variables. You can find derivations in e.g. Baley/Veldkamp (2021): Bayesian Learning
See also the related answer at
...
- 479
3
votes
Limit of random walk auto correlation function
Hi: The expression tends to 1.0. The intuition is that, as $t$ gets larger and larger, the $h$ lags that seperate the two processes become more and more negligible and the processes begin to look the ...
- 486
3
votes
Application of Poisson process in economic modelling
Klette and Kortum (2004) develop a parsimonious model of innovating firms rich enough to confront firm-level evidence. It captures the dynamics of individual heterogenous firms, describes the behavior ...
- 6,885
3
votes
Accepted
Purpose of Semidefinite Integral
It is just an opaque way to denote that the non-definite integration limit will take the value of the variable itself. Namely,
$$\int_a g(x) dx \equiv \int_a^x g(s)ds$$
... where $s$ is just the ...
- 33.4k
2
votes
Accepted
Profit maximization under uncertainity
Let $G$ be the cumulative distribution function for the buyers' willingness to pay. If there is no cost, expected profit is simply expected revenue. The revenue is quantity times price. If the unit of ...
- 11.7k
2
votes
Given $\mathbb Q$ and $X_t$ is $\mathbb Q$-Brownian, find $\frac{d\mathbb Q}{d\mathbb P}$ / Uniqueness of Brownian or Radon-Nikodym derivative
As you say (how did $\sin t$ in the initial problem become $\cos t$?), $\mathbb{Q}$ is a measure under which $W_t$ becomes $\tilde{W}_t - \sin t$, where $\tilde{W}_t$ is a $\mathbb{Q}$-Brownian motion....
- 2,619
2
votes
Accepted
Decomposition of an additive functional into a Martingale part and other
Using the given formulas (inserting the expression for the $X$'s in the sum), we arrive at
$$Y_t = (1-\beta_0)\alpha_0\cdot t + \beta_0\sum_{j=0}^{t-1}X_j +\sum_{j=1}^{t}W_j $$
We do not need a ...
- 33.4k
2
votes
Accepted
Are overlapping generation (OLG) models extensions of a DSGE model?
You can find OLG models that do not classify as DSGE (in particular, the model might not be stochastic) as well as DSGE with overlapping generations (contrary to those with infinitely lived agents).
...
- 862
2
votes
Accepted
Is playing against state of nature considered Stochastic game or Bayesian game?
To me, Bayesian games are usually "static" in the sense that no "time" is involved. The state is drawn once, and players make decisions, possibly sequentially, and the outcome is ...
- 15.3k
2
votes
Accepted
Bond Price expression
If $ \delta > 0 $ is very small, then the interest incurred during the small subperiod $[t, t + \delta]$ can be approximated using the simple interest formula. More specifically, the interest ...
- 460
1
vote
Granger-Sims causality and subtle differences
Let $\mathbf{X}$ be conditionally iid on $\{0,1\}$ with distribution $(P,1-P)$, with $P$ being a nontrivial random variable. Let $Y_t=\limsup_{T\to\infty} T^{-1}\sum_{i=1}^TX_{t-i}$.
Then $\textbf{X}\...
- 11.7k
1
vote
Stochastic optimal control problem (calculus)
I think there should be a minus before the final term in the expression you are looking for. In any case, you did not plug in the optimality condition for $w$, which is in your case
$$
w = -\frac{V'(A)...
1
vote
Derivation of autocovariances Lewis (2021) RES
A useful implication of conditional mean independence is: $E[\varepsilon_{it}|\varepsilon_{ks}]=0 \implies E[\varepsilon_{it}\varepsilon_{ks}]=0,$ and more generally, $E[\varepsilon_{it}|\varepsilon_{...
- 3,144
1
vote
Accepted
Expectational stability: adaptive learning of RE equilibria in dynamic systems
These are many questions. O.k., so let's go step by step:
(Q1) What is a mapping actually?
A map is just another term for a function. Here, every "law of motion", the actual one (ALM) and the ...
- 6,550
1
vote
Autocorrelation function of a random walk process
By recursive substitution you obtain $y_t = \sum_{j=0}^t e_{t-j}.$ Thus $\forall p \in \mathbb{N} \hspace{.2cm} y_{t-p} = \sum_{j=0}^{t-p} e_{t-p-j}.$
Under the usual white noise assumption for the ...
- 1,278
1
vote
Decomposition of an additive functional into a Martingale part and other
Usually additive functionals are defined for (strong) Markov processes with continuous sample paths (diffusions) but I suppose you do have a Markov---AR(1)---time series and $\{ Y_t \}$ is indeed ...
- 2,619
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