# Expectation VS forecast

I'm confused at the difference between the 'expectation' and 'forecast'. In behavioral economics, forecast bias is defined as the difference between expectation and forecast. However both sound pretty similar to me. What's the difference conceptually?

• Some professional forecasters will use "we forecast.." and "we expect.." interchangeably. Jan 12 at 14:08

To avoid confusion inherent in colloquial expressions, it is convenient to define and analyze expectations and forecasts using mathematics and statistics.

An expectation is then the expected value $$\mathbb{E}(X)$$ of an underlying random variable $$X$$.
It may be unconditional or conditional, usually the latter, and we can make that explicit: $$\mathbb{E}(X|I)$$ where $$I$$ is the conditioning variable(s) which usually represent the available information (e.g. a sample of data).
In a time series setting, the expected value can also be indexed by time such as $$\mathbb{E}(X_{t+h}|I_t)$$ where we have the expectation of the variable $$X$$ for $$h^{th}$$ period ahead from $$t$$ conditional on the information $$I$$ available at $$t$$.
An expectation can also be true or just estimated: $$\mathbb{E}(\cdot)$$ vs. $$\hat{\mathbb{E}}(\cdot)$$. Usually, only the latter is available to us.

A forecast is a guess of a yet unrealized or yet unknown value $$x$$ of an underlying random variable $$X$$. There are point forecasts that consist of a single point $$\hat x$$, interval forecasts $$(\hat x_{\text{lower}},\hat x_{\text{upper}})$$ or an entire density forecast stating the density of $$X$$ as $$\hat f_X(\cdot)$$.

A relationship: Under square loss (quadratic loss), an unconditionally optimal point forecast of $$x$$ is $$\mathbb{E}(X)$$. This is usually unavailable, so we tend to use its empirical counterpart $$\hat{\mathbb{E}}(X)$$ instead. And we often have some information we can condition on, so we end up with $$\hat{\mathbb{E}}(X|I)$$ or $$\hat{\mathbb{E}}(X_{t+h}|I_t)$$. However, the expected value is not generally optimal under other relevant loss functions such as absolute loss, quantile loss (a.k.a. tick or pinball loss) and other. Under these loss functions, other functions of the distribution of $$X$$ (characterized by $$f_X(\cdot)$$) are optimal point forecasts, and it makes sense to use their sample counterparts when the true values are not available.

One is still free to use the word expectation in its nonmathematical sense, but in any case, it is best to be explicit about it so that no one gets confused. (My personal experience is that the discrepancy between the mathematical meaning and the other meanings do get conflated both in academia and in the private sector and that causes real problems, at least in communication if not beyond.)

• Thanks a lot! Sounds clear! Jan 19 at 3:21
• @user14261785, you are welcome! If the answer is helpful and clear, you may eventually accept it by clicking on the tick mark to the left. Otherwise, you may ask for further clarification. This is how Economics SE works. Jan 19 at 5:56

When you roll a dice the expectation is $$3.5$$, if you always forecast $$3$$, your forecast bias will be $$0.5$$.

If the inflation on average rises by $$3\%$$, but you predict that it rises by $$2\%$$, your forecast bias is $$1pp$$.

We usually cannot know the real expectation and, hence, the real forecast bias. We can only estimate it by, for example, using the average of past forecast errors. (Forecast error is the difference between our prediction and the actual realization of the variable of interest.)

• Do you mean that forecast is coming from our 'hunch' and expectation's coming from expectation forming formula? Jan 12 at 19:47
• Forecast comes from your model. Expectation = expected value, it is a property of the phenomenon you try to model. Jan 13 at 6:15

Forecasts are based on data that's tangible prior to the event you are forecasting. Forecasts also are for a specific day in the future. Expectations can also be based on past data however with expectations the person who is 'expecting' 100 units of XYZ can also manipulate or create or do or not do something that would cause the expectation to be what it is. For example you expect 10 people at a party......you are having a say in that right? Yes. Your party planner however they need to provide an accurate forecast so they can only monitor the rsvp and make a forecast.

• I (not a native English speaker) have never heard of the distinction in the second part of this answer, about the forecaster being unable to affect events. Can you please support this claim with a reference? Jan 10 at 7:55
• Thanks for your comment, but still it doesn't resolve my concern completely. Let's take an example of macro variables. What about 'forecast of inflation' vs 'expectation of inflation'? Forecasts are just results of survey? Jan 10 at 17:09