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1# The rolling-time-dummy (RTD) method — definitions from Hill, Scholz, Shimizu & Steurer23**Target citation.** Hill, R. J., Scholz, M., Shimizu, C. & Steurer, M. (2022). "Rolling-Time-Dummy House4Price Indexes: Window Length, Linking and Options for Dealing with Low Transaction Volume." *Journal of5Official Statistics* 38(1), 127–151. DOI 10.2478/jos-2022-0007 (`hill2022rtd`).67**Access caveat (important).** On 2026-10-21 the published JOS text could not be retrieved by any8automated route (SAGE `journals.sagepub.com` and `reference-global.com` return HTTP 403 for both the HTML9and the PDF endpoints; Sciendo redirects to reference-global; no mirror found). Only the **abstract** of10the 2022 article was obtained (Crossref). The definitions below are therefore taken from the **working-paper11version**: Hill, Scholz, Shimizu & Steurer (2020), "Rolling-Time-Dummy House Price Indexes: Window Length,12Linking and Options for Dealing with the Covid-19 Shutdown", Graz Economics Papers 2020-14 (August 2020),1338 pp., PDF at `https://www.csis.u-tokyo.ac.jp/wp-content/uploads/2020/08/166.pdf` (CSIS Discussion Paper14166; RePEc `grz:wpaper:2020-14`). **Page references below are to that working paper, not to the JOS15pagination.** The JOS abstract confirms the same three contributions (window-length criterion, linking16variants, three modifications for low transaction volume), and Ishaak et al. (2024, p. 395) cite the JOS17version for "a method to determine the optimal length of a window". Anything that may have changed between18the 2020 WP and the 2022 article (numerical results, terminology, thresholds) is marked **[WP only — verify19against JOS]**.2021---2223## 1. The standard RTD method (WP §2.1, pp. 2–3)2425Defined "with a window length of k + 1 periods, as defined in Shimizu et al. (2010) and O'Hanlon (2011)".26With period t the first period in the window, estimate by OLS the semilog hedonic model, Equation (1), p. 3:2728 ln p_h = Σ_{c=1}^{C} β_c z_{hc} + Σ_{s=t+1}^{t+k} δ_s d_{hs} + ε_{hs}2930where h indexes transactions falling in the window, z_{hc} are characteristics, d_{hs} = 1 if the31dwelling sold in period s (zero otherwise), ε iid mean zero. **Note: in the window the hedonic32coefficients β are common to all k + 1 periods (one pooled regression), and the base period t has no33dummy (P_t = 1 within the window).**3435The index change for the **last pair of periods only** is taken from the window, Equation (2), p. 3:3637 P_{t+k} / P_{t+k−1} = exp(δ̂^t_{t+k}) / exp(δ̂^t_{t+k−1})3839(superscript t = base period of the window's regression). Then "The window is then rolled forward one40period and the hedonic model is re-estimated", giving Equation (3): P_{t+k+1}/P_{t+k} =41exp(δ̂^{t+1}_{t+k+1}) / exp(δ̂^{t+1}_{t+k}). The multi-period index is the chain of these bilateral links,42Equation (4), p. 3.4344Key property (p. 3): "once a price change P_{t+k}/P_{t+k−1} has been computed, it is never revised. ...45The sole objective when re-estimating the hedonic model to include period t + k + 1 is to compute46P_{t+k+1}/P_{t+k}."4748**Terminology link (WP p. 5):** this standard link "is sometimes referred to as a **movement splice** (see49for example de Haan 2015, and Chessa, Verburg and Willenborg 2017)". Ishaak et al. (2024, p. 402) compute50their RTD benchmark exactly this way: "window length of three periods and a movement splice as described51by De Haan (2015)".5253## 2. Linking variants (WP §2.2, pp. 3–5)5455Given a window of k + 1 periods, the new period can be linked to any earlier period in the window:5657- Link to **two periods earlier**, Equation (5), p. 4:58 P_{t+k}/P_{t+k−1} = (P_{t+k−2}/P_{t+k−1}) · exp(δ̂^t_{t+k}) / exp(δ̂^t_{t+k−2}),59 where P_{t+k−1}, P_{t+k−2} are already fixed.60- Link to **three periods earlier**, Equation (6), and in general to **j periods earlier (j ≤ k)**,61 Equation (7), p. 4:62 P_{t+k}/P_{t+k−1} = (P_{t+k−j}/P_{t+k−1}) · exp(δ̂^t_{t+k}) / exp(δ̂^t_{t+k−j}).63 "given a window length of k + 1 periods, there are k distinct ways of linking period t + k with the64 earlier periods. Each will give a different answer, and one cannot say ex ante that one is better than65 another" (p. 4).66- **Geometric mean of the k single-period links**, Equation (8), p. 4:67 P_{t+k}/P_{t+k−1} = [ Π_{j=1}^{k} (P_{t+k−j}/P_{t+k−1}) · exp(δ̂^t_{t+k})/exp(δ̂^t_{t+k−j}) ]^{1/k}.68 The WP (p. 5) says this "is analogous to" the **mean splice** of the scanner-data literature "(see69 Diewert and Fox 2017)".70- **Weighted geometric mean** with geometrically declining weights, Equation (9), p. 4: weights71 (1−λ)λ^{j−1}/(1−λ^k), 0 < λ < 1, more weight on recent links; compared (p. 5) to Melser's (2018)72 weighted mean splice, with the caveat that Melser's weights come from product overlaps, "No such73 equivalent weights exist in our context."74- **Window splice** is mentioned (p. 5) as the scanner-data method "which links the new month in by75 comparing it with the corresponding month one year earlier (see Krsinich 2014)"; it is **not** adopted76 as an RTD variant in the WP. (Note: in Ishaak et al. 2024, "window splice" denotes something different —77 their Step 4 geometric average over base-period series.)78- **Half splice:** the term **does not appear** in the 2020 WP. Whether the JOS 2022 version uses it is79 **unverified**. Do not attribute a "half splice" definition to Hill et al. without checking the JOS text.8081## 3. Low-transaction-volume modifications (WP §2.3, pp. 5–9; JOS title: "Options for Dealing with Low Transaction Volume")8283Motivation (p. 5): a period with very few transactions has a noisy δ̂; "This will have a permanent impact84on the RTD price index for all subsequent periods, which will now be subject to upward [or downward]85drift." Starting point: "the desired window length is known, and that special action is deemed necessary86for any period that has less than N transactions" (p. 6). **The threshold N is left to the user in the87method description; in the Sydney illustration (Figure 14, p. 29) weeks with fewer than 250 transactions88are treated as low-volume [WP only — verify N in JOS].**8990- **Method 1 (p. 6–7):** when computing P_{t+k}, any earlier period in the window with < N transactions is91 dropped from the window and replaced by "the most recent available earlier period that has at least N92 transactions". If period t+k itself has < N, its index P_{t+k}/P_{t+k−1} is still computed "(or93 P_{t+k}/P_{t+k−2} if period t+k−1 has less than N transactions)", "But period t + k is then not used to94 compute the price indexes of later periods." Worked example with a 3-month window and months 4–6 thin:95 months 4, 5, 6, 7 are all linked via month 3; "From then on normal chronological chaining ... resumes"96 (p. 6, Figure 1 p. 7).97- **Method 2 (p. 7–8):** a period with < N transactions is supplemented with the chronologically closest98 transactions from the preceding period until N is reached ("If say there are enough transactions in the99 last 10 days of the preceding period ... only these last 10 days are added"); those transactions remain100 in the previous period too. Then the standard RTD is applied.101- **Method 3 (p. 8):** no index is computed for a period with < N; it is merged with the next period102 (and further periods) until the merged period reaches N, then treated as a single period.103- Choice guidance (p. 8–9): "If it is important that a price index is computed for every period ... then104 either method 1 or 2 should be used. Method 1 will better capture actual price movements during the105 shutdown unless it becomes too distorted by noise arising from small sample sizes." Figure 14 (p. 29):106 "Failure to adjust for low transaction weeks seems to cause a slight upward drift in the RTD index."107108## 4. The window-length / linking-method selection criterion (WP §3.3, pp. 11–13)109110The criterion compares a high-frequency (weekly) RTD index with a **reference lower-frequency (quarterly)111hedonic index** — two references are used: quarterly hedonic imputation and quarterly time dummy (§3.1–3.2),112chosen "because they are quite different (i.e., one re-estimates the hedonic model every quarter while the113other does not re-estimate at all)" (p. 11). Footnote 4: "With slight modifications, the same method can114be used to determine the optimal window length for monthly indexes as well."115116Step 1 — convert the weekly index to quarterly, Equation (17), p. 12 (unweighted; a weighted variant (18)117gives "very similar" results):118119 P^w_{t,t+1} = [ Π_{v=1}^{13} P_{t+1,v} / P_{t,v} ]^{1/13}120121(P_{t,v} = weekly index level in quarter t, week v; 14 weeks occasionally).122123Step 2 — distance to the reference quarterly index with Diewert's (2002, 2009) metrics, p. 13:124125 X1 = (1/(T−1)) Σ_t [ P^w/P^quart + P^quart/P^w − 2 ]126 X2 = (1/(T−1)) Σ_t { [P^w/P^quart − 1]² + [P^quart/P^w − 1]² }127128"The smaller the value of the X metric, the more similar are the two indexes" (p. 13); rankings by X1 and129X2 "are almost identical", so X1 is reported. "We prefer whichever window length generates the smallest X130metric" and, holding the window fixed, "the linking method with the smallest X metric" (p. 13).131132## 5. Recommended window length and linking (WP §5 and §6, pp. 21–30) [WP only — verify numbers against JOS]133134- Weekly indexes are "quite sensitive to the choice of window length" (window varied 2–53 weeks, p. 21);135 sensitivity to the linking method is smaller but present (p. 21).136- **Optimal window length: Sydney 19–21 weeks (21 with the hedonic-imputation reference, 18 with the137 time-dummy reference, 19 with geometric linking); Tokyo about 18 weeks** (pp. 22–24, conclusion p. 29).138- **Optimal linking: for Sydney, link the current week with a period 13–16 weeks earlier; for Tokyo, 12139 weeks earlier. "Geometric averages of the single-period linking methods performs about equally well as140 the best of the single-period linking methods"** (conclusion pp. 29–30).141- The WP gives **no general-purpose recommended window length** (e.g. for monthly municipal indexes); it142 gives a procedure (Section 4 above) to be run on each dataset, plus the empirical answers for weekly143 Sydney and Tokyo. For context, Ishaak et al. (2024, p. 395) summarise the JOS version as: "a longer window144 generally increases the robustness of the index, while a shorter window increases the current market145 relevance."146147## 6. What the JOS 2022 abstract (verified) confirms148149"Rolling-time-dummy (RTD) is a hedonic method used by a number of countries to compute their official house150price indexes (HPIs). The RTD method requires less data and is more adaptable than other hedonic methods,151which makes it well suited for computing higher frequency HPIs (e.g., monthly or weekly). In this article,152we address three key issues relating to RTD. First, we develop a method for determining the optimal length153of the rolling window. Second, we consider variants on the standard way of linking the current period with154earlier periods, and show how the optimal linking method can be determined. Third, we propose three ways155of modifying the RTD method to make it more robust to periods of low transaction volume."156157## 7. Reference implementation in the CBS `REPS` package (for benchmark D)158159`REPS::calculate_rolling_timedummy()` (R/calculate_rolling_timedummy.R, REPS 1.1.1): fits a pooled time-dummy160regression on the first `window_length` periods, converts the within-window index to growth rates, then for161each subsequent window keeps **only the last growth rate** of the new window and chains it (`cumprod`) —162i.e. the standard movement splice of Section 1. Default `window_length = 5` if not supplied163(`calculate_hedonic_index.R`). No alternative linking (j-period, mean, weighted mean) and no low-volume164methods are implemented in the package.165166## 8. Not in the sources (do not guess)167168- JOS 2022 page numbers for each definition; any wording changes from "Covid-19 Methods 1–3" to the JOS169 "options for low transaction volume"; whether the JOS version adds results for monthly data or a170 threshold N other than the Sydney illustration's 250/week.171- "Half splice": absent from the WP; status in JOS unknown.172- A universal recommended window length: none given; only the selection criterion and the weekly173 Sydney/Tokyo optima.174