# The rolling-time-dummy (RTD) method — definitions from Hill, Scholz, Shimizu & Steurer

Target citation. Hill, R. J., Scholz, M., Shimizu, C. & Steurer, M. (2022). "Rolling-Time-Dummy House Price Indexes: Window Length, Linking and Options for Dealing with Low Transaction Volume." Journal of Official Statistics 38(1), 127–151. DOI 10.2478/jos-2022-0007 (hill2022rtd).

Access caveat (important). On 2026-10-21 the published JOS text could not be retrieved by any automated route (SAGE journals.sagepub.com and reference-global.com return HTTP 403 for both the HTML and the PDF endpoints; Sciendo redirects to reference-global; no mirror found). Only the abstract of the 2022 article was obtained (Crossref). The definitions below are therefore taken from the working-paper version: Hill, Scholz, Shimizu & Steurer (2020), "Rolling-Time-Dummy House Price Indexes: Window Length, Linking and Options for Dealing with the Covid-19 Shutdown", Graz Economics Papers 2020-14 (August 2020), 38 pp., PDF at https://www.csis.u-tokyo.ac.jp/wp-content/uploads/2020/08/166.pdf (CSIS Discussion Paper 166; RePEc grz:wpaper:2020-14). Page references below are to that working paper, not to the JOS pagination. The JOS abstract confirms the same three contributions (window-length criterion, linking variants, three modifications for low transaction volume), and Ishaak et al. (2024, p. 395) cite the JOS version for "a method to determine the optimal length of a window". Anything that may have changed between the 2020 WP and the 2022 article (numerical results, terminology, thresholds) is marked [WP only — verify against JOS].

# 1. The standard RTD method (WP §2.1, pp. 2–3)

Defined "with a window length of k + 1 periods, as defined in Shimizu et al. (2010) and O'Hanlon (2011)". With period t the first period in the window, estimate by OLS the semilog hedonic model, Equation (1), p. 3:

ln p_h = Σ_{c=1}^{C} β_c z_{hc} + Σ_{s=t+1}^{t+k} δ_s d_{hs} + ε_{hs}

where h indexes transactions falling in the window, z_{hc} are characteristics, d_{hs} = 1 if the dwelling sold in period s (zero otherwise), ε iid mean zero. Note: in the window the hedonic coefficients β are common to all k + 1 periods (one pooled regression), and the base period t has no dummy (P_t = 1 within the window).

The index change for the last pair of periods only is taken from the window, Equation (2), p. 3:

P_{t+k} / P_{t+k−1} = exp(δ̂^t_{t+k}) / exp(δ̂^t_{t+k−1})

(superscript t = base period of the window's regression). Then "The window is then rolled forward one period and the hedonic model is re-estimated", giving Equation (3): P_{t+k+1}/P_{t+k} = exp(δ̂^{t+1}{t+k+1}) / exp(δ̂^{t+1}{t+k}). The multi-period index is the chain of these bilateral links, Equation (4), p. 3.

Key property (p. 3): "once a price change P_{t+k}/P_{t+k−1} has been computed, it is never revised. ... The sole objective when re-estimating the hedonic model to include period t + k + 1 is to compute P_{t+k+1}/P_{t+k}."

Terminology link (WP p. 5): this standard link "is sometimes referred to as a movement splice (see for example de Haan 2015, and Chessa, Verburg and Willenborg 2017)". Ishaak et al. (2024, p. 402) compute their RTD benchmark exactly this way: "window length of three periods and a movement splice as described by De Haan (2015)".

# 2. Linking variants (WP §2.2, pp. 3–5)

Given a window of k + 1 periods, the new period can be linked to any earlier period in the window:

Link to two periods earlier, Equation (5), p. 4: 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}), where P_{t+k−1}, P_{t+k−2} are already fixed.
Link to three periods earlier, Equation (6), and in general to j periods earlier (j ≤ k), Equation (7), p. 4: 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}). "given a window length of k + 1 periods, there are k distinct ways of linking period t + k with the earlier periods. Each will give a different answer, and one cannot say ex ante that one is better than another" (p. 4).
Geometric mean of the k single-period links, Equation (8), p. 4: 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}. The WP (p. 5) says this "is analogous to" the mean splice of the scanner-data literature "(see Diewert and Fox 2017)".
Weighted geometric mean with geometrically declining weights, Equation (9), p. 4: weights (1−λ)λ^{j−1}/(1−λ^k), 0 < λ < 1, more weight on recent links; compared (p. 5) to Melser's (2018) weighted mean splice, with the caveat that Melser's weights come from product overlaps, "No such equivalent weights exist in our context."
Window splice is mentioned (p. 5) as the scanner-data method "which links the new month in by comparing it with the corresponding month one year earlier (see Krsinich 2014)"; it is not adopted as an RTD variant in the WP. (Note: in Ishaak et al. 2024, "window splice" denotes something different — their Step 4 geometric average over base-period series.)
Half splice: the term does not appear in the 2020 WP. Whether the JOS 2022 version uses it is unverified. Do not attribute a "half splice" definition to Hill et al. without checking the JOS text.

# 3. Low-transaction-volume modifications (WP §2.3, pp. 5–9; JOS title: "Options for Dealing with Low Transaction Volume")

Motivation (p. 5): a period with very few transactions has a noisy δ̂; "This will have a permanent impact on the RTD price index for all subsequent periods, which will now be subject to upward [or downward] drift." Starting point: "the desired window length is known, and that special action is deemed necessary for any period that has less than N transactions" (p. 6). The threshold N is left to the user in the method description; in the Sydney illustration (Figure 14, p. 29) weeks with fewer than 250 transactions are treated as low-volume [WP only — verify N in JOS].

Method 1 (p. 6–7): when computing P_{t+k}, any earlier period in the window with < N transactions is dropped from the window and replaced by "the most recent available earlier period that has at least N transactions". If period t+k itself has < N, its index P_{t+k}/P_{t+k−1} is still computed "(or 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 to compute the price indexes of later periods." Worked example with a 3-month window and months 4–6 thin: months 4, 5, 6, 7 are all linked via month 3; "From then on normal chronological chaining ... resumes" (p. 6, Figure 1 p. 7).
Method 2 (p. 7–8): a period with < N transactions is supplemented with the chronologically closest transactions from the preceding period until N is reached ("If say there are enough transactions in the last 10 days of the preceding period ... only these last 10 days are added"); those transactions remain in the previous period too. Then the standard RTD is applied.
Method 3 (p. 8): no index is computed for a period with < N; it is merged with the next period (and further periods) until the merged period reaches N, then treated as a single period.
Choice guidance (p. 8–9): "If it is important that a price index is computed for every period ... then either method 1 or 2 should be used. Method 1 will better capture actual price movements during the shutdown unless it becomes too distorted by noise arising from small sample sizes." Figure 14 (p. 29): "Failure to adjust for low transaction weeks seems to cause a slight upward drift in the RTD index."

# 4. The window-length / linking-method selection criterion (WP §3.3, pp. 11–13)

The criterion compares a high-frequency (weekly) RTD index with a reference lower-frequency (quarterly) hedonic index — two references are used: quarterly hedonic imputation and quarterly time dummy (§3.1–3.2), chosen "because they are quite different (i.e., one re-estimates the hedonic model every quarter while the other does not re-estimate at all)" (p. 11). Footnote 4: "With slight modifications, the same method can be used to determine the optimal window length for monthly indexes as well."

Step 1 — convert the weekly index to quarterly, Equation (17), p. 12 (unweighted; a weighted variant (18) gives "very similar" results):

P^w_{t,t+1} = [ Π_{v=1}^{13} P_{t+1,v} / P_{t,v} ]^{1/13}

(P_{t,v} = weekly index level in quarter t, week v; 14 weeks occasionally).

Step 2 — distance to the reference quarterly index with Diewert's (2002, 2009) metrics, p. 13:

X1 = (1/(T−1)) Σ_t [ P^w/P^quart + P^quart/P^w − 2 ]
X2 = (1/(T−1)) Σ_t { [P^w/P^quart − 1]² + [P^quart/P^w − 1]² }

"The smaller the value of the X metric, the more similar are the two indexes" (p. 13); rankings by X1 and X2 "are almost identical", so X1 is reported. "We prefer whichever window length generates the smallest X metric" and, holding the window fixed, "the linking method with the smallest X metric" (p. 13).

# 5. Recommended window length and linking (WP §5 and §6, pp. 21–30) [WP only — verify numbers against JOS]

Weekly indexes are "quite sensitive to the choice of window length" (window varied 2–53 weeks, p. 21); sensitivity to the linking method is smaller but present (p. 21).
Optimal window length: Sydney 19–21 weeks (21 with the hedonic-imputation reference, 18 with the time-dummy reference, 19 with geometric linking); Tokyo about 18 weeks (pp. 22–24, conclusion p. 29).
Optimal linking: for Sydney, link the current week with a period 13–16 weeks earlier; for Tokyo, 12 weeks earlier. "Geometric averages of the single-period linking methods performs about equally well as the best of the single-period linking methods" (conclusion pp. 29–30).
The WP gives no general-purpose recommended window length (e.g. for monthly municipal indexes); it gives a procedure (Section 4 above) to be run on each dataset, plus the empirical answers for weekly Sydney and Tokyo. For context, Ishaak et al. (2024, p. 395) summarise the JOS version as: "a longer window generally increases the robustness of the index, while a shorter window increases the current market relevance."

# 6. What the JOS 2022 abstract (verified) confirms

"Rolling-time-dummy (RTD) is a hedonic method used by a number of countries to compute their official house price indexes (HPIs). The RTD method requires less data and is more adaptable than other hedonic methods, which makes it well suited for computing higher frequency HPIs (e.g., monthly or weekly). In this article, we address three key issues relating to RTD. First, we develop a method for determining the optimal length of the rolling window. Second, we consider variants on the standard way of linking the current period with earlier periods, and show how the optimal linking method can be determined. Third, we propose three ways of modifying the RTD method to make it more robust to periods of low transaction volume."

# 7. Reference implementation in the CBS `REPS` package (for benchmark D)

REPS::calculate_rolling_timedummy() (R/calculate_rolling_timedummy.R, REPS 1.1.1): fits a pooled time-dummy regression on the first window_length periods, converts the within-window index to growth rates, then for each subsequent window keeps only the last growth rate of the new window and chains it (cumprod) — i.e. the standard movement splice of Section 1. Default window_length = 5 if not supplied (calculate_hedonic_index.R). No alternative linking (j-period, mean, weighted mean) and no low-volume methods are implemented in the package.

# 8. Not in the sources (do not guess)

JOS 2022 page numbers for each definition; any wording changes from "Covid-19 Methods 1–3" to the JOS "options for low transaction volume"; whether the JOS version adds results for monthly data or a threshold N other than the Sydney illustration's 250/week.
"Half splice": absent from the WP; status in JOS unknown.
A universal recommended window length: none given; only the selection criterion and the weekly Sydney/Tokyo optima.