seeking advice on a series

[newsagent]

I have a monthly series on deposit flows (dep_hh)

kk1<-structure(list(dep_hh = c(1295.19, 638.889999999999, 1137.74000000001, 
416.929999999995, 2892.00000000001, 2309.78999999999, 1273.67000000001, 
1339.54, 76.2499999999977, -589.089999999999, -375.770000000004, 
4259.95, -2605.79, 1249.18, -254.67, 95.1199999999967, 364.799999999996, 
552.380000000006, 21.909999999998, 1023.02, 1339.03, -712.109999999997, 
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679.257621605933, -557.309925199958, -199.673082468486, 443.575010406266, 
-1984.5484232557, -9203.48264797664, -5386.52202997756, -2427.60418385226, 
-3289.75029411654, -3080.66331408241, -7200.0555807555, -1332.02965160322, 
-937.178410076676, -199.437373413972, -532.756593955326, -530.372308507822, 
1452.16527451732, 188.320059670931, -706.933572511068, -58.9561349405675, 
759.874739996891, -840.398510718998, 368.200911363775, 241.630388684751, 
278.031178856834, -31.1465115531427, 656.606312211385, -433.991856143106, 
2175.82044463047, -972.327270247884, -636.998109795661, -312.204039998699, 
443.941941928059, -213.520519050771, 770.666177993155, 415.600807102909, 
691.435107115522, 189.792300176782, 584.817214937239, 66.5951944719479, 
2631.84260629687, -1060.0283486877, 505.302406819899, 992.012403223878, 
3.92684103861759, 339.367041357146, 1086.03628239419, 539.693269632847, 
763.983597827145, 384.183590391225, -78.1336186027216, 60.2907964526892, 
2287.06490901851, -538.6933703161, 277.0247256181, 516.0253692618, 
832.5711401348, 344.9621281137, 1075.0518373961, 561.7028982283, 
1043.2274721938, 101.2438352752, 456.9832627264, 87.4852865186999, 
1846.9909846013, -966.9719678628, 457.3620906008, 1340.7853660751, 
1520.8384707545, 165.2531138499, 112.0916121104, 669.5612891303, 
165.4064623325, 282.7341938597, 1558.6104824552, 1284.1481936342, 
3432.6574892398, 671.0371277249, 304.3945761819, 792.3677211723, 
1809.1172016544, 77.5076130477, 542.4993618865, 786.5578549725, 
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986.1252669008, -335.5945939238, 721.6263081758, 787.7868033505, 
464.679237009, 191.7442455131, 121.1953234953, -408.5511658739, 
3213.2832020411, -1418.9992659473, -201.5750262328, 737.632523723, 
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347.6012596841, -274.1918057196, 775.4806139965, 173.159947921, 
1034.8729370933, 329.7923586171, -136.5063920524, 551.2391797012, 
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349.9354489763, 355.7272373446, 696.575432726, 742.3321953899, 
1008.0008475498, -104.5437542846, 195.5767867012, 941.1958908976, 
2470.6343636759), month2 = structure(c(11323, 11354, 11382, 11413, 
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20209, 20240, 20270, 20301, 20332, 20362, 20393, 20423), class = "Date"), 
    yemo = structure(c(11323, 11354, 11382, 11413, 11443, 11474, 
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"tbl", "data.frame")), index = structure("yemo", ordered = TRUE), index2 = "yemo", interval = structure(list(
    year = 0, quarter = 0, month = 1, week = 0, day = 0, hour = 0, 
    minute = 0, second = 0, millisecond = 0, microsecond = 0, 
    nanosecond = 0, unit = 0), .regular = TRUE, class = c("interval", 
"vctrs_rcrd", "vctrs_vctr")), row.names = c(NA, -300L), class = c("tbl_ts", 
"tbl_df", "tbl", "data.frame"))

Some test results:

kk1 %>%   select(yemo,dep_hh) %>%  
 features(dep_hh , c(unitroot_kpss,unitroot_pp, unitroot_ndiffs, unitroot_nsdiffs))

# A tibble: 1 × 6
  kpss_stat kpss_pvalue pp_stat pp_pvalue ndiffs nsdiffs
      <dbl>       <dbl>   <dbl>     <dbl>  <int>   <int>
1     0.375      0.0880   -12.8      0.01      0       0

Using an ARIMA I get

kk1 %>% model(ARIMA(dep_hh))
# A mable: 1 x 1
            `ARIMA(dep_hh)`
                    <model>
1 <ARIMA(1,0,3)(2,0,0)[12]>

The arima seems to produce reasonable forecasts
I also take an OLS model given by

                                 `diff(dep_hh) =a+b*dlog(unemployment)+e_t  `

where diff() is the first difference of dep_hh

The reason for using diff(dep_hh) is that when I take an adverse scenario for unemployment the forecasted dep_hh changes reasonably compared to the the forecasts obtained from the baseline scenario about unemployment
But If I take

                                    `dep_hh =a+b*dlog(unemployment)+e_t  `

then when I take the same adverse scenario for unemployment the forecasted dep_hh remain essentially the same and very very close to the forecasts obtained from the baseline scenario about unemployment, which is not reasonable.

My question is: can I use diff of dep_hh in my OLS model despite the fact that I used dep_hh in my ARIMA, or should I use only dep_hh across all models?

[robjhyndman]

The issue here is that you probably want to forecast dep_hh, not diff(dep_hh). So if you fit a model to the latter, and produce forecasts, you then have to undo the differencing, and it is difficult to produce a distributional forecast.

Assuming you want to forecast dep_hh, then write all the models in terms of dep_hh. It is possible to write your OLS model on differences, as a dynamic regression model with ARIMA(0,1,0) errors:

kk1 |>
  model(
    arima = ARIMA(dep_hh),
    ols = ARIMA(dep_hh ~ log(unemployment) + pdq(0,1,0) + PDQ(0,0,0))
  )

Other potential models involving covariates can be written in the same way. You possibly don't want to specify an ARIMA(0,1,0) here, I'm just showing you how to produce an equivalent model to your OLS model. A full dynamic regression model with automatically selected ARIMA errors can be estimated by dropping the pdq and PDQ terms.

Don't worry about the fact that dep_hh is non-stationary. ARIMA will take care of that in the estimation process (another advantage of using ARIMA over a straight regression).

[newsagent]

Many thanks! Is this series non-stationary? The selected ARIMA does not take the first difference of the series.
If it is non-stationary and has seasonality, should we remove the seasonality before testing for unit root?