Evidence above strongly points to KOI-774.01 showing two distinct overlapping periodicities in its TTV plot (see especially Figure 3. above).  It seems possible that the longer periodicity  (~ 1504 to ~1754 days) could be due to another unseen planet with a somewhat larger period (or, as suggested by Dan Fabrycky, precession of the planetary orbit) while the shorter periodicity (~ 31.5 days) could be due to (1) another unseen planet relatively nearby, or (2) a large exomoon.
For the latter case, in the absence of other prominent perturbers (as in this case), the time between “transits” of the planet-moon barycenter should be constant.  Since an orbiting moon should continuously alter the position of the barycenter with respect to Kepler’s line-of-sight, while the center of the star’s light-flux drop will predominantly be controlled by the large transiting planetary disc irrespective of the barycenter’s position, a TTV would be expected.  While the period of such an exomoon that could lead to a P_ttv of ~ 31.5 days is, of course, unknown, as a point of interest, the 8 largest moons in our Solar System (Earth’s Moon, Europa, Io, Ganymede, Callisto, Titan, Titania, and Triton) have periods ranging from 1.8 to 27.3 days.
These observations are similar to those disclosed for the KOI-139.02 and -676.02 systems (see web pages).
It would be interesting and exciting to see how (or if) orbital mechanical calculations could distinguish between and/or favor (1) or (2).
KOI-774 is a very noisy star.  Based on the Lomb-Scargle periodogram of the original Kepler light-curve data and what appears to be repeating (albeit gradually altering) starspot arrays, the star’s rotation rate is 10.41 days; this LSP peak had a huge Power of 6159.58 and a P-value of 0.
One other test of the data was carried out at the suggestion of Eric Ford.  Plotting just the Out-Of-Transit (OOT) data points to 1 day on either side of each transit gave curved arrays that could be well-defined by quadratic equations.  The derivative of each of these equation was evaluated at the corresponding mid-transit time to give an estimate of the OOT-slope-at-mid-transit.  These slopes were then plotted against the corresponding (O - C)'s to unEarth any unusual correlation that might be present.  None was found; the plot was just a random scatter of points.  This would then seem to rule out the star's noisy (starspot) behavior as being involved in the 31.5 day periodicity.


Linear ephemeris (this work): Tc = 7.44265221(Tc#) + 102.97203082

Figure 1.: KOI-774.01, P = 7.44 days [Plot error bars (smaller than symbol) = ± 0.63 min.]
[Predicted: TTV_minimum: -84.01 ± 0.48 days, Amp_ttv_minimum: -0.68 ± 0.35 min]
OBSERVED:    TTV_maximum: 776.17 ± 0.48 days, Amp_ttv_maximum: 0.47 ± 0.35 min.
[Predicted: TTV_minimum: 1636.34 ± 0.48 days, Amp_ttv_minimum: -0.68 ± 0.35 min.]
P_ttv: 1720.35* ± 15.42 days.
Amp_ttv: 1.15 ± 0.49 minutes.
Same data: Lomb-Scargle periodogram, two P_ttv candidates:
1504.66 days; Power: 16.15; P-value: 0.000171; and
31.51 days; Power: 13.53; P-value: 0.00235;
no other credible peaks were present.

Figure 2.: KOI-774.01
TTV_maximum: 74.53 ± 0.12 days, Amp_ttv_maximum: 0.44 ± 0.10 min.
TTV_minimum: 90.28 ± 0.12 days, Amp_ttv_minimum: -0.48 ± 0.10 min.
TTV_maximum: 106.03 ± 0.12 days, Amp_ttv_maximum: 0.44 ± 0.10 min.
TTV_minimum: 121.77 ± 0.12 days, Amp_ttv_minimum: -0.48 ± 0.10 min.
     (+41 other intervening maxima & minima)
TTV_maximum: 1428.86 ± 0.12 days, Amp_ttv_maximum: 0.44 ± 0.10 min.
TTV_minimum: 1444.61 ± 0.12 days, Amp_ttv_minimum: -0.48 ± 0.10 min.
TTV_maximum: 1460.36 ± 0.12 days, Amp_ttv_maximum: 0.44 ± 0.10 min.
TTV_minimum: 1476.11 ± 0.12 days, Amp_ttv_minimum: -0.48 ± 0.10 min.
P_ttv: 31.50 ± 0.01 days.
Amp_ttv: 0.91 ± 0.14 minutes.
Same data: Lomb-Scargle periodogram, two P_ttv candidates (see above).


Figure 3.: KOI-774.01
Arithmetic sum of the curves of Figure 1. and Figure 2. showing the system's dual oscillation reality.

Figure 4.: KOI-774.01
TTV_maximum: 74.50 ± 0.11 days, Amp_ttv_maximum: 0.44 ± 0.09 min.
TTV_minimum: 90.24 ± 0.11 days, Amp_ttv_minimum: -0.48 ± 0.09 min.
TTV_maximum: 105.99 ± 0.11 days, Amp_ttv_maximum: 0.44 ± 0.09 min.
TTV_minimum: 121.74 ± 0.11 days, Amp_ttv_minimum: -0.48 ± 0.09 min.
     (+ 41 other intervening maxima & minima)
TTV_maximum: 1428.83 ± 0.11 days, Amp_ttv_maximum: 0.44 ± 0.09 min.
TTV_minimum: 1444.58 ± 0.11 days, Amp_ttv_minimum: -0.48 ± 0.09 min.
TTV_maximum: 1460.32 ± 0.11 days, Amp_ttv_maximum: 0.44 ± 0.09 min.
TTV_minimum: 1476.07 ± 0.11 days, Amp_ttv_minimum: -0.48 ± 0.09 min.
P_ttv: 31.50 ± 0.01 days.
Amp_ttv: 0.92 ± 0.13 minutes.
Same data: Lomb-Scargle periodogram, one P_ttv candidate:
31.48 days; Power: 16.70; P-value: 0.00000993;
no other credible peaks were present.

Figure 5.: KOI-774.01
[Predicted: TTV_minimum: -103.43 ± 0.46 days, Amp_ttv_minimum: -0.71 ± 0.35 min]
OBSERVED: TTV_maximum: 773.46 ± 0.46 days, Amp_ttv_maximum: 0.47 ± 0.35 min.
[Predicted: TTV_minimum: 1650.35 ± 0.46 days, Amp_ttv_minimum: -0.71 ± 0.35 min.]
P_ttv: 1753.79* ± 14.74 days.
Amp_ttv: 1.18 ± 0.49 minutes.
Same data: Lomb-Scargle periodogram, one P_ttv candidate:
1504.66 days; Power: 19.18; P-value: 0.00000824;
no other credible peaks were present.
==========================================================================
Duration, tau, & Depth vs. Transit Time
Kepler KOI-774 (KIC-11656840) 1-(or more)-Planet(s)

TTVs are derived from Q1-Q16 Kepler data.  x-axes: “Observed Tc” (Mid-Transit Time): EXOFAST’s best-fits from Kepler light flux vs. time data.  y-axes: “(O – C)”: difference between Observed Tc and the Calculated Tc from the graphically obtained linear ephemeris.  The plots are pictured in the order of orbital periods.
EXOFAST data were obtained using the recently (Aug'13) updated value for [Fe/H] of -0.308 as one of the Priors.  Dual oscillations were found for KOI-774.01.  The Lomb-Scargle Periodogram (LSP) of the TTV x-y data ((O-C) vs. Time) for KOI-774.01 showed two prominent periodicities both of which appeared to be highly credible due to their low P-values (i.e., "by-chance" probabilities):
LSP-Periodicity-A: 1504.66 days, Power: 16.15, P-value: 0.000171; and
LSP-Periodicity-B: 31.51 days, Power: 13.53; P-value: 0.00235;
the same data gave the following two sinusoidal "best-fit" curves:
Sinus.-A: 1720.35* ± 15.42 days, Average Amplitude: 0.57 ± 0.35 min.; and
Sinus.-B: 31.50 ± 0.01 days, Average Amplitude: 0.46 ± 0.10 min.
Residuals-of-Sinus.-A (differences between the initial set of data points and those calculated from the sinusoidal equation defining Sinus.-A) showed one prominent, highly credible LSP periodicity:
LSP-Periodicity-C: 31.48 days, Power: 16.69, P-value: 0.00000993;
along with the following sinusoidal "best-fit" curve:
Sinus.-C: 31.50 ± 0.01 days, Average Amplitude: 0.46 ± 0.09 min.
Note the close agreement between LSP-Periodicity-B & LSP-Periodicity-C, and between Sinus.-B & Sinus.-C.
An LSP of a (secondary) set of residuals determined from Sinus.-C showed no prominent periodicities at all.  Also, Residuals-of-Sinus.-B (differences between the initial set of data points and those calculated from the sinusoidal equation defining Sinus.-B) showed one prominent, highly credible LSP periodicity:
LSP-Periodicity-D: 1504.66 days, Power: 19.18, P-value: 0.00000824;
along with the following sinusoidal "best-fit" curve:
Sinus.-D: 1753.79* ± 14.74 days, Average Amplitude: 0.59 ± 0.35 min.
Note the close agreement between LSP-Periodicity-A & LSP-Periodicity-D, and between Sinus.-A & Sinus.-D.
An LSP of a (secondary) set of residuals determined from Sinus.-D showed no prominent periodicities at all.
* These values are derived from the best-fit sinusoidal equations which, unfortunately, were mostly based upon a single very broad maximum.  Based on a large number of other systems, better agreement with the Lomb-Scargle periodogram periodicities would be expected if more data were available to generate a more accurate sinusoidal equation.The results are graphically arrayed below in the same order they were discussed.

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Duration, tau, & Depth vs. Transit Time
[Within EXOFAST's program, "tau" signifies [(Ingress time) + (Egress time)].


Duration vs. Transit Time
LSP-Periodicity-E: 31.37 days, Power: 10.36, P-value: 0.0541; no other credible ones.
This periodicity is the same as one of the TTV's observed earlier.
Sinus.-E: 31.35 ± 0.01 days; Amplitude varies between 0.0968 to 0.1005 (± 0.0005) days.
Figure 6. (above) shows that most values for duration near 0.0975 days with excursions (up to ~ 0.1157 days) only to higher values.

tau vs. Transit Time
LSP-Periodicity-F: 31.37 days, Power: 11.24, P-value: 0.0229; no other credible ones.
Again, this periodicity is the same as one of the TTV's observed earlier.
Sinus.-F: 31.36 ± 0.01 days; Amplitude varies between 0.0127 to 0.0171 (± 0.0005) days.
Figure 7. (above) shows that most of the values of tau appear near 0.0125 days with excursions (up to ~ 0.0302 days) only to higher values.

Depth vs. Transit Time
LSP-Periodicity-G: 380.11 days, Power: 13.39, P-value: 0.00270; no other credible ones.
Sinus.-G: 379.71 ± 0.52 days; Amplitude varies between 0.0220 to 0.0236 (± 0.0002) (decimal).
Figure 8. (above) shows these results.  This is likely just a seasonal effect of the Kepler telescope since the periodicity is very similar to that of Kepler's orbit, namely ~ 372.5 days.
The best-fit sinusoidal data are:
TTV_minimum: 154.26 ± 5.04 days, Amp_ttv_minimum: 0.0220 ± 0.0002
TTV_maximum: 344.11 ± 5.04 days, Amp_ttv_maximum: 0.0236 ± 0.0002
TTV_minimum: 533.97 ± 5.04 days, Amp_ttv_minimum: 0.0220 ± 0.0002
TTV_maximum: 723.82 ± 5.04 days, Amp_ttv_maximum: 0.0236 ± 0.0002
TTV_minimum: 913.68 ± 5.04 days, Amp_ttv_minimum: 0.0220 ± 0.0002
TTV_maximum: 1103.53 ± 5.04 days, Amp_ttv_maximum: 0.0236 ± 0.0002
TTV_minimum: 1293.39 ± 5.04 days, Amp_ttv_minimum: 0.0220 ± 0.0002
TTV_maximum: 1483.24 ± 5.04 days, Amp_ttv_maximum: 0.0236 ± 0.0002

References:
• For Q1-Q6 TTV data of 774.01 (thru ~ 563 (BJD-2454900) days), see: Ford2012arXiv-1201.1892.

30 Aug 2013; updated 4 Sep 2013