**UPDATE:** Further communication with the BoE clarified that the spreadsheet corresponding to the August Monetary Policy Report does not contain the parameter $\gamma$ itself but the spread between the mean and the mode of the forecast distribution. To be consistent with this approach, I released a new version of my fanchart package. See details here: New Release of fanchart package. I thank the BoE for providing further details on their data. The take-home messages from this post stay the same: (1) I would be beneficial to get more details on the estimation of the parameters (2) Skewness matters! đź™‚

The August Monetary Policy Report was recently published by the Bank of England (BoE). As usual, it contains their customary fancharts showing the forecast distributions for CPI inflation.

To make these plots, the BoE assumes that forecast for CPI Inflation follows a twopiece normal distribution (you can find further details on the BoE methodology here). More precisely, that the probability density function (pdf) of the CPI inflation forecast is given by $$

s(x) := s\left(x; \mu,\sigma,\gamma\right) =

\begin{cases}

\dfrac{A}{\sqrt{2\pi}\sigma} \exp \left\{ -\dfrac{1-\gamma}{2\sigma^2} \left[(x-\mu)^2\right] \right\}, \qquad \mbox{if } x < \mu, \\

\dfrac{A}{\sqrt{2\pi}\sigma} \exp \left\{ -\dfrac{1+\gamma}{2\sigma^2} \left[(x-\mu)^2\right] \right\}, \qquad \mbox{if } x \geq \mu, \\

\end{cases}

$$ where $A = \dfrac{2}{\frac{1}{\sqrt{1-\gamma}}+\frac{1}{\sqrt{1+\gamma}}}$, and there are three parameters that need to be estimated:

- $\mu \in \mathbb{R}$ : Mode – a measure of the central tendency for inflation
- $\sigma>0$ : Uncertainty – a view on the degree of dispersion
- $\gamma\in (-1,1)$ : Skewness – a view on the balance of the risks

Once these parameters are set, the forecast is completely defined. The BoE publishes its estimated parameters as part of their charts and data quarterly package (See the Downloads sections in Monetary Poly Report August). This quarter, the following table of parameters was published.

Noticeably, the skew parameters $\gamma$ for 2023-Q2 and 20223-Q3 (in both Market and 1.75% interest rate scenarios) are outside of the range $(-1, 1)$ with values 1.08 and 1.0, respectively. These values are not valid since evaluating them in the expression

$$A = \dfrac{2}{\frac{1}{\sqrt{1-\gamma}}+\frac{1}{\sqrt{1+\gamma}}},$$

would produce an imaginary number for 1.08 and a division by zero for 1.0.

## Take-Home Messages

First, the need for more information about the methodology that the BoE uses to find the parameters estimates. Making this information available would allow anybody to understand and reproduce the estimates provided by the BoE. This would increase transparency and confidence in the forecasts and ultimately in the BoE.

Second, the skewness parameter is becoming more relevant reflecting the importance of the choice of the twopiece distribution for modelling CPI inflation:

- In November 2021, the estimates for the skewness was equal to zero for all forecast periods (see image below). This implies that the CPI inflation forecasts follow a normal distribution (since the twopiece normal becomes a simple normal/Gaussian when $\gamma=0$).
- However, since February 2022, we have mostly positive skewness estimates. This implies a greater probability of the CPI inflation being above the mode rather than below it, over the corresponding periods. Note that this can be interpreted as a more negative forecast, since higher levels of inflations are more likely.
- In the latest report August 2022, the skewness estimates corresponding to 2024-Q2 to 2025-Q1 are negative. This would imply that the probability of CPI inflation being below the mode is greater than that of being above it.

This illustrates the impact of the skewness parameter in the forecasts.

- If the skewness is zero, then we have a symmetrical forecast. As we mentioned before, the forecast follows a normal distribution in such case. So, observing inflation above the central projection (note that in this case the mode, median, and mean are all the same) is equally likely than observing inflation below the central projection.
- However, if the skewness is not zero, then its sign tells us the more likely direction of the CPI inflation forecast. For example in the August 2022 report the BoE projects:
- No skewness, i.e. symmetric forecasts, until the end of the year
- Positive skewness for the whole 2023 and the first quarter of 2024. That is, the probability of observing inflation above the central projection is bigger than that of observing inflation below it. Note that this can be interpreted as a more negative forecast since higher levels of inflations are more likely.
- Negative skewness from 2024-Q2 until 2025-Q1. Here, the BoE projects greater probability of observing inflation below the central projection than above it.
- No skewness in the 2025-Q2 forecasts
- A small positive skewness in the 2025-Q3 forecast

After these take home messages, I will wait for the Bank of England to respond with corrected parameters or a clarification on their methodology, and then produce some new fancharts.