`boiwsa`

is an R package for seasonal adjustment of weekly data. It provides a simple, easy-to-use interface for calculating the seasonally adjusted estimates, as well as a number of diagnostic tools for evaluating the quality of the adjustments.

The seasonal adjustment procedure approach aligns closely with the locally-weighted least squares procedure introduced by Cleveland et al. (2014) albeit with several adjustments. Firstly, instead of relying on differencing to detrend the data, we opt for a more explicit approach by directly estimating the trend through smoothing techniques. Secondly, we incorporate a variation of Discount weighted regression (Harrison and Johnston, 1984) to enable the seasonal component to evolve dynamically over time.

We consider the following decomposition model of the observed series \(y_{t}\):

\[ y_{t}=T_{t}+S_{t}+H_{t}+O_{t}+I_{t}, \]

where \(T_{t}\), \(S_{t}\), \(H_{t}\), \(O_{t}\) and \(I_{t}\) represent the trend, seasonal, outlier, holiday- and trading-day, and irregular components, respectively. To evade ambiguity, it is important to note that in weekly data, \(t\) typically denotes the date of the last day within a given week. The seasonal component is specified using trigonometric variables as:

\[ \begin{eqnarray*} S_{t} &=&\sum_{k=1}^{K}\left( \alpha _{k}^{y}\sin (\frac{2\pi kD_{t}^{y}}{ n_{t}^{y}})+\beta _{k}^{y}\cos (\frac{2\pi kD_{t}^{y}}{n_{t}^{y}})\right) + \\ &&\sum_{l=1}^{L}\left( \alpha _{l}^{m}\sin (\frac{2\pi lD_{t}^{m}}{n_{t}^{m}} )+\beta _{l}^{m}\cos (\frac{2\pi lD_{t}^{m}}{n_{t}^{m}})\right) , \end{eqnarray*} \]

where \(D_{t}^{y}\) and \(D_{t}^{m}\) are the day of the year and the day of the month, and \(n_{t}^{y}\) and \(n_{t}^{m}\) are the number of days in the given year or month. Thus, the seasonal adjustment procedure takes into account the existence of two cycles, namely intrayearly and intramonthly.

Like the X-11 method (Ladiray and Quenneville, 2001), the `boiwsa`

procedure uses an iterative principle to estimate the various components. The seasonal adjustment algorithm comprises eight steps, which are documented below:

Step 1: Estimation of trend (\(T_{t}^{(1)}\)) using

`stats::supsmu()`

.Step 2: Estimation of the Seasonal-Irregular component:

\[y_{t}-T_{t}^{(1)}=S_{t}+H_{t}+O_{t}+I_{t}\]

Step 2*: Searching for additive outliers using the method proposed by Finfley et al. (1998)

Step 2**: Identifying the optimal number of trigonometric variables

Step 3: Calculation of seasonal factors, along with other potential factors such as \(H_{t}\) or \(O_{t}\), is achieved through discount weighted regression on the seasonal-irregular component extracted in Step 2. In this application, the discounting rate decays over the years. For each year \(t\) and the observed year \(\tau\), a geometrically decaying weight function is represented as: \(w_{t}=r^{|t-\tau|}\), where \(r \in (0,1]\). This approach differs from the customary one-way discounting, allowing us to incorporate future observations in computing seasonal factors and thus avoid the limitations of the forecasting methods mentioned in Bandara et al. (2021). In addition, in traditional discount-weighted regression, even with a conservative choice of \(r=0.95\), in weekly data, observations separated by more than \(2\) years would carry nearly negligible weight. Thus, the use of year-based discounting prevents an overly rapid decay which may potentially lead to unstable estimates of the seasonal component.

- Step 4: Estimation of trend (\(T_{t}^{(2)}\)) from seasonally and outlier adjusted series using
`stats::supsmu()`

function (R Core Team, Step 5: Estimation of the Seasonal-Irregular component: \[y_{t}-T_{t}^{(2)}=S_{t}+H_{t}+O_{t}+I_{t}\]

Step 6: Computing the final seasonal factors (and possibly other factors such as \(H_{t}\) or \(O_{t}\)) using discount weighted regression, as in step 3.

Step 7: Estimation of the final seasonally adjusted series: \[y_{t}-S_{t}-H_{t}\]

Step 8: Computing final trend (\(T_{t}^{(3)}\)) estimate from seasonally and outlier adjusted series using

`stats::supsmu()`

.

To install boiwsa, you can use devtools:

Alternatively, you can clone the repository and install the package from source:

Using `boiwsa`

is simple. First, load the `boiwsa`

package:

Next, load your time series data into a data frame object. Here is an example that is based on the `gasoline`

data from the `fpp2`

package:

```
data("gasoline.data")
plot(gasoline.data$date,gasoline.data$y,type="l",xlab="Year",ylab=" ", main="Weekly US gasoline production")
```

Once you have your data loaded, you can use the `boiwsa`

function to perform weekly seasonal adjustment:

The `x`

argument takes the series to be seasonally adjusted, while the dates argument takes the associated dates in date format. Unless specified otherwise (i.e., `my.k_l = NULL`

), the procedure automatically identifies the best number of trigonometric variables to capture the yearly (\(K\)) and monthly (\(L\)) cycles based on the AICc. The information criterion is specified by the `ic`

option. The weighting decay rate is specified by `r`

(by default `r=0.8`

).

The procedure automatically searches for additive outliers (AO) using the method described in Appendix C of Findley et al. (1998). To disable the automatic AO search, set `auto.ao.search = F`

. To add user-defined AOs, use the `ao.list`

option.

The `boiwsa`

function returns a list object containing the results. The seasonally adjusted series is stored in a vector called `sa`

. In addition, the estimated seasonal factors are stored as `sf`

.

You can then plot the adjusted data to visualize the seasonal pattern:

```
plot(gasoline.data$date,gasoline.data$y,type="l",xlab="Year",ylab=" ", main="Weekly US gasoline production")
lines(gasoline.data$date,res$sa,col="red")
legend(
"topleft",
legend = c("Original", "SA"),
lwd = c(2,2),
col = c("black", "red"),
bty = "n"
)
```

To evaluate the quality of the adjustment, you can use the `plot_spec`

function provided by the package, which generates a plot of the autoregressive spectrum of the raw and seasonally adjusted data:

Bandara, K., Hyndman, R. J. and C. Bergmeir (2021). MSTL: A seasonal-trend decomposition algorithm for time series with multiple seasonal patterns. arXiv preprint arXiv:2107.13462 .

Cleveland, W.P., Evans, T.D. and S. Scott (2014). Weekly Seasonal Adjustment-A Locally-weighted Regression Approach (No. 473). Bureau of Labor Statistics.

Findley, D.F., Monsell, B.C., Bell, W.R., Otto, M.C. and B.C Chen (1998). New capabilities and methods of the X-12-ARIMA seasonal-adjustment program. Journal of Business & Economic Statistics, 16(2), pp.127-152.

Harrison, P. J. and F. R. Johnston (1984). Discount weighted regression. Journal of the Operational Research Society 35(10), 923–932.

Hyndman, R. (2023). fpp2: Data for “Forecasting: Principles and Practice” (2nd Edition). R package version 2.5.

Ladiray, D. and B. Quenneville (2001). Seasonal adjustment with the X-11 method.

R Core Team (2013). R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. ISBN 3-900051-07-0.

The views expressed here are solely of the author and do not necessarily represent the views of the Bank of Israel.

Please note that `boiwsa`

is still under development and may contain bugs or other issues that have not yet been resolved. While we have made every effort to ensure that the package is functional and reliable, we cannot guarantee its performance in all situations.

We strongly advise that you regularly check for updates and install any new versions that become available, as these may contain important bug fixes and other improvements. By using this package, you acknowledge and accept that it is provided on an “as is” basis, and that we make no warranties or representations regarding its suitability for your specific needs or purposes.