Part 1. A Small Economy with a Large Seasonal Pulse
Montenegro’s GDP as a quarterly time series
Gross domestic product is the broadest single measure of macroeconomic activity. It summarises the value of goods and services produced in an economy and therefore provides a compact view of expansion, contraction, recovery and longer-term changes in productive capacity. In this blog post, Montenegro’s GDP is analysed as a quarterly index, with 2021=100, covering the period from 2006Q1 to 2026Q1. The data source is Eurostat’s quarterly national accounts dataset, namq_10_gdp. Two versions of the series are examined: the original series and the seasonally adjusted series obtained using X13-ARIMA.
Figure 1 immediately shows that Montenegro’s GDP has a very pronounced seasonal pattern. The original transformed series moves in a regular saw-tooth shape, with strong peaks and troughs inside each year. The seasonally adjusted series therefore plays a particularly important role. Without seasonal adjustment, it would be easy to confuse regular quarterly movement with genuine changes in macroeconomic momentum.
The long-run picture is one of growth, but with unusually large cyclical interruptions. The early part of the sample shows a strong rise up to the pre-global-financial-crisis period. After that, the seasonally adjusted series becomes flatter and more uneven. The 2009–2012 period looks like one of adjustment rather than dynamic expansion. From the mid-2010s to 2019, the series resumes a clearer upward path. The COVID-19 shock in 2020 is the dominant event in the graph: GDP falls sharply and visibly, far more dramatically than in most ordinary business-cycle episodes. The rebound that follows is also strong. By the later part of the sample, GDP is again above its pre-pandemic level, but the final observations suggest some moderation around a high level rather than a new acceleration.
This pattern is consistent with the structure of a small, open and highly seasonal economy. The graph itself does not identify the sectoral cause of the movement, but the size of the quarterly seasonal swing and the dramatic pandemic interruption are exactly the kind of pattern one expects in an economy where seasonal services and external demand matter strongly. For analytical purposes, this makes Montenegro a particularly interesting case: the statistical tools must handle both trend growth and strong seasonality, while also recognising that some shocks are too large to be treated as normal cyclical noise.
Figures 2 and 3 show that the persistence of Montenegro’s GDP is more seasonal than purely quarter-to-quarter. The first autocorrelation of the log-level series is about 0.360, which is lower than in some smoother macroeconomic GDP series. However, the seasonal autocorrelation is high, about 0.857, and the autocorrelation graph shows large spikes at seasonal lags. This means that the same quarter of the year contains a lot of information about the expected level of GDP. In other words, Montenegro’s GDP is not only persistent over time; it is strongly organised by the calendar.
Figure 3 shows what happens after first and seasonal differencing of the log series. The first autocorrelation of the differenced series is close to zero, around 0.013, while the second-difference autocorrelation is slightly negative. The differenced autocorrelation pattern still contains a noticeable seasonal imprint, but the dominant level and seasonal persistence have been substantially reduced. This supports the use of differencing before ARIMA modelling.
Table 1 confirms the visual evidence. Trend strength is high, about 0.966, though not as close to one as in some larger and smoother economies. This indicates that the long-run component is important, but that cyclical and seasonal movements are also large. Seasonal strength is extremely high, about 0.978. The seasonal peak is in the third quarter and the trough is in the first quarter, which matches the graph. The spikiness measure is extremely low, meaning that the series is not simply a collection of isolated random spikes. Its movements are structured, even when they are large.
Table 1. Features of GDP for Montenegro time series
| Feature | Value |
|---|---|
| c | 0.966 |
| seasonal_strength_year | 0.978 |
| seasonal_peak_year | 3 |
| seasonal_trough_year | 1 |
| spikiness | 0.000 |
| linearity | 1.243 |
| curvature | 0.191 |
| stl_e_acf1 | -0.120 |
| stl_e_acf10 | 0.423 |
| acf1 | 0.360 |
| acf10 | 1.940 |
| diff1_acf1 | 0.013 |
| diff1_acf10 | 3.867 |
| diff2_acf1 | -0.014 |
| diff2_acf10 | 3.809 |
| season_acf1 | 0.857 |
| pacf5 | 1.206 |
| diff1_pacf5 | 0.997 |
| diff2_pacf5 | 1.094 |
| season_pacf | 0.355 |
| zero_run_mean | 0.000 |
| nonzero_squared_cv | 0.003 |
| zero_start_prop | 0.000 |
| zero_end_prop | 0.000 |
| lambda_guerrero | 2.000 |
| kpss_stat | 1.929 |
| kpss_pvalue | 0.010 |
| pp_stat | -6.094 |
| pp_pvalue | 0.010 |
| ndiffs | 1 |
| nsdiffs | 1 |
| bp_stat | 10.470 |
| bp_pvalue | 0.001 |
| lb_stat | 10.862 |
| lb_pvalue | 0.001 |
| var_tiled_var | 0.075 |
| var_tiled_mean | 0.364 |
| shift_level_max | 0.378 |
| shift_level_index | 2 |
| shift_var_max | 0.067 |
| shift_var_index | 59 |
| shift_kl_max | 0.575 |
| shift_kl_index | 55 |
| spectral_entropy | 0.488 |
| n_crossing_points | 35 |
| longest_flat_spot | 2 |
| coef_hurst | 0.755 |
| stat_arch_lm | 0.773 |
The linearity measure is positive, and curvature is also positive. This suggests a rising long-run path with changes in slope rather than a simple straight-line trend. The STL residual autocorrelation is negative at the first lag but positive over longer lags, which means that some structure remains after decomposition. The ACF and PACF features show moderate short-run dependence but strong seasonal dependence. The zero-run and zero-proportion measures are zero, as expected for a continuous GDP index. The Guerrero lambda is close to two, suggesting that the chosen transformation context should be interpreted carefully; nevertheless, the log transformation is still useful for modelling proportional movement in a highly seasonal series.
The stationarity diagnostics indicate that the level series should not be treated as stationary. The KPSS statistic is high, with a p-value at the lower bound, and the automatic differencing indicators recommend one ordinary difference and one seasonal difference. The Box-Pierce and Ljung-Box statistics are highly significant in the level series, confirming serial dependence. The shift-level indicator points to a major early-sample level shift, while the shift-variance and shift-distribution measures point to important changes around the later part of the sample, consistent with the pandemic and post-pandemic period. Spectral entropy is moderate, around 0.488, implying a series that is structured but also more complex than the smoother GDP paths seen in some neighbouring economies. The Hurst coefficient, about 0.755, indicates persistence, though less extreme than in very smooth trending series. The ARCH statistic is not large enough to make volatility clustering the central feature.
The conclusion from Part 1 is that Montenegro’s GDP is a strongly seasonal, trending and structurally uneven quarterly series. It is shaped by a rising long-run path, powerful intra-year seasonality, a severe pandemic shock and a strong rebound. These features explain why seasonal adjustment, transformation and differencing are essential before forecasting.
Methodological appendix to Part 1
Graphical exploration is the first stage of time-series analysis. It helps identify trend, seasonality, cyclical movements, structural breaks, volatility changes and outliers. For Montenegro, the most obvious visual feature is the strong quarterly seasonal pattern. The main risk is to mistake normal seasonal variation for a genuine shift in economic direction.
A log transformation is often useful when macroeconomic fluctuations become larger in absolute terms as the level of the series rises. Logs make movements easier to interpret as approximate percentage changes. First differences focus on short-run change, while seasonal differences compare a quarter with the same quarter in the previous year and help remove recurring seasonal structure.
The features in Table 1 summarise the statistical behaviour of the series. Trend strength, seasonal strength, autocorrelation, stationarity diagnostics, entropy, spikiness, shift measures and persistence indicators jointly show whether the series is trending, seasonal, persistent or structurally unstable. For Montenegro, the dominant message is strong seasonality combined with meaningful trend and large cyclical disturbance.
