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# 金融代写|金融计量经济学代考FINANCIAL ECONOMETRICS代考|ECON345 Parameter-Centric Analysis Grossly Exaggerates Certainty

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## 金融代写|金融计量经济学代考FINANCIAL ECONOMETRICS代考|Over-Certainty

Suppose an observable $y \sim \operatorname{Normal}(0,1)$; i.e., we characterize the uncertainty in an observable $y$ with a normal distribution with known parameters (never mind how we know them). Obviously, we do not know with exactness what any future value of $y$ will be, but we can state probabilities (of intervals) for future observables using this model.

It might seem an odd way of stating it, but in a very real sense we are infinitely more certain about the value of the model parameters than we are about values of the observable. We are certain of the parameters’ values, but we have uncertainty in the observable. In other words, we know what the parameters are, but we don’t know what values the observable will take. If the amount of uncertainty has any kind of measure, it would be 0 for the value of the parameters in this model, and something positive for the value of the observable. The ratio of these uncertainties, observable to parameters, would be infinite.

That trivial deduction is the proof that, at least for this model, certainty in model parameters is not equivalent to certainty in values of the observable. It would be an obvious gaff, not even worth mentioning, were somebody to report uncertainty in the parameters as if it were the same as the uncertainty in the observable.

Alas, this is what is routinely done in probability models, see Chap. 10 of Briggs (2016). Open the journal of almost any sociology or economics journal, and you will find the mistake being made everywhere. If predictive analysis were used instead of parameteric or testing-based analysis, this mistake would disappear; see e.g. Ando (2007), Arjas and Andreev (2000), Berkhof and van Mechelen (2000), Clarke and Clarke (2018). And then some measure of sanity would return to those fields which are used to broadcasting “novel” results based on statistical model parameters.
The techniques to be described do not work for all probability models; only those models where the parameters are “like” the observables in a sense to be described.

## 金融代写|金融计量经济学代考FINANCIAL ECONOMETRICS代考|Theory

There are several candidates for a measure of total uncertainty in a proposition. Since all probability is conditional, this measure will be, too. A common measure is variance; another is the length of the highest (credible) density interval. And there are more, such as entropy, which although attractive has a limitation described in the final section. I prefer here the length of credible intervals because they are stated in predictive terms in many models in units of the observable, made using plainlanguage probability statements. Example: “There is a $90 \%$ chance $y$ is in $(a, b)$.”
In the $y \sim \operatorname{Normal}(0,1)$ example, the variance of the uncertainty of either parameter is 0 , as is the length of any kind of probability interval around them. The variance of the observable is 1 , and the length of the $1-\alpha$ density interval around the observable $y$ is well known to be $2 z_{\alpha / 2}$, where $z_{\alpha / 2} \approx 2$. The ratio of variances, parameter to observable, is $0 / 1=0$. The ratio of the length of confidence intervals, here observable to parameter, is $4 / 0=\infty$.

We pick the ratio of the length of the $1-\alpha$ credible intervals as observable to parameter to indicate the amount of over-certainty. If not otherwise indicated, I let $\alpha$ equal the magic number.

In the simple Normal example, as said in the beginning, if somebody were to make the mistake of claiming the uncertainty in the observable was identical to the uncertainty of the parameters, he would be making the worst possible mistake. Naturally, in situations like this, few or none would this blunder.

Things change, though, and for no good reason, when there exists or enters uncertainty in the parameter. In these cases, the mistake of confusing kinds of uncertainty happens frequently, almost to the point of exclusively.
The simplest models with parameter uncertainty follow this schema:
$$p(y \mid \mathrm{DB})=\int_{\theta} p(y \mid \theta, \mathrm{DB}) p(\theta \mid \mathrm{DB}) d \theta$$
where $\mathrm{D}=y_{1}, \ldots, y_{n}$ represents old measured or assumed values of the observable, and B represents the background information that insisted on the model formulation used. D need not be present. B must always be; it will contain the reasoning for the model form $p(y \mid \theta \mathrm{DB})$, the form of the model of the uncertainty in the parameters $p(\theta \mid \mathrm{DB})$, and the values of hyperparameters, if any. Obviously, if there are two (or more) contenders $i$ and $j$ for priors on the parameters, then in general $p\left(y \mid \mathrm{DB}{k}\right) \neq p\left(y \mid \mathrm{DB}{l}\right)$. And if there are two (or more) sets of D, $k$ and $l$, then in general $p\left(y \mid \mathrm{D}{i} \mathrm{~B}\right) \neq p\left(y \mid \mathrm{D}{j} \mathrm{~B}\right)$. Both $\mathrm{D}$ and $\mathrm{B}$ may differ simultaneously, too.

## 金融代写|金融计量经济学代考 FINANCIAL ECONOMETRICS代 考|Theory

$$p(y \mid \mathrm{DB})=\int_{\theta} p(y \mid \theta, \mathrm{DB}) p(\theta \mid \mathrm{DB}) d \theta$$

## MATLAB代写

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